diff --git a/changelog b/changelog index c467da5..6b98774 100644 --- a/changelog +++ b/changelog @@ -1,3 +1,14 @@ +20080330 tpd src/input/Makefile add integration regression testing +20080330 tpd src/input/schaum11.input integrals of sqrt(a^2-x^2) +20080330 tpd src/input/schaum10.input integrals of sqrt(x^2-a^2) +20080330 tpd src/input/schaum9.input integrals of sqrt(x^2+a^2) +20080330 tpd src/input/schaum8.input integrals of a^2-x^2, x^2a^2 +20080330 tpd src/input/schaum6.input make regression testing uniform +20080330 tpd src/input/schaum5.input make regression testing uniform +20080330 tpd src/input/schaum4.input make regression testing uniform +20080330 tpd src/input/schaum3.input make regression testing uniform +20080330 tpd src/input/schaum2.input make regression testing uniform 20080328 tpd src/input/Makefile add integration regression testing 20080328 tpd src/input/schaum6.input integrals of x^2+a^2 20080328 tpd src/input/schaum5.input integrals of sqrt(ax+b) and sqrt(px+q) diff --git a/src/input/Makefile.pamphlet b/src/input/Makefile.pamphlet index d4caed2..a1382e4 100644 --- a/src/input/Makefile.pamphlet +++ b/src/input/Makefile.pamphlet @@ -356,7 +356,8 @@ REGRES= algaggr.regress algbrbf.regress algfacob.regress alist.regress \ realclos.regress reclos.regress repa6.regress robidoux.regress \ roman.regress roots.regress ruleset.regress rules.regress \ schaum1.regress schaum2.regress schaum3.regress schaum4.regress \ - schaum5.regress schaum6.regress \ + schaum5.regress schaum6.regress schaum7.regress schaum8.regress \ + schaum9.regress schaum10.regress schaum11.regress \ scherk.regress scope.regress seccsc.regress \ segbind.regress seg.regress \ series2.regress series.regress sersolve.regress set.regress \ @@ -630,7 +631,9 @@ FILES= ${OUT}/algaggr.input ${OUT}/algbrbf.input ${OUT}/algfacob.input \ ${OUT}/robidoux.input ${OUT}/roman.input ${OUT}/roots.input \ ${OUT}/ruleset.input ${OUT}/rules.input ${OUT}/schaum1.input \ ${OUT}/schaum2.input ${OUT}/schaum3.input ${OUT}/schaum4.input \ - ${OUT}/schaum5.input ${OUT}/schaum6.input \ + ${OUT}/schaum5.input ${OUT}/schaum6.input ${OUT}/schaum7.input \ + ${OUT}/schaum8.input ${OUT}/schaum9.input ${OUT}/schaum10.input \ + ${OUT}/schaum11.input \ ${OUT}/saddle.input \ ${OUT}/scherk.input ${OUT}/scope.input ${OUT}/seccsc.input \ ${OUT}/segbind.input ${OUT}/seg.input ${OUT}/series2.input \ @@ -932,6 +935,9 @@ DOCFILES= \ ${DOC}/schaum1.input.dvi ${DOC}/schaum2.input.dvi \ ${DOC}/schaum3.input.dvi ${DOC}/schaum4.input.dvi \ ${DOC}/schaum5.input.dvi ${DOC}/schaum6.input.dvi \ + ${DOC}/schaum7.input.dvi ${DOC}/schaum8.input.dvi \ + ${DOC}/schaum9.input.dvi ${DOC}/schaum10.input.dvi \ + ${DOC}/schaum11.input.dvi \ ${DOC}/s01eaf.input.dvi ${DOC}/s13aaf.input.dvi \ ${DOC}/s13acf.input.dvi ${DOC}/s13adf.input.dvi \ ${DOC}/s14aaf.input.dvi ${DOC}/s14abf.input.dvi \ diff --git a/src/input/schaum10.input.pamphlet b/src/input/schaum10.input.pamphlet new file mode 100644 index 0000000..8582086 --- /dev/null +++ b/src/input/schaum10.input.pamphlet @@ -0,0 +1,757 @@ +\documentclass{article} +\usepackage{axiom} +\begin{document} +\title{\$SPAD/input schaum10.input} +\author{Timothy Daly} +\maketitle +\eject +\tableofcontents +\eject +\section{\cite{1}:14.210~~~~~$\displaystyle\int{\frac{dx}{\sqrt{x^2-a^2}}}$} +$$\int{\frac{1}{\sqrt{x^2-a^2}}}=\ln\left(x+\sqrt{x^2-a^2}\right)$$ +<<*>>= +)spool schaum10.output +)set message test on +)set message auto off +)clear all + +--S 1 of 28 +aa:=integrate(1/(sqrt(x^2-a^2)),x) +--R +--R +--R +-------+ +--R | 2 2 +--R (1) - log(\|x - a - x) +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.xxx~~~~~$\displaystyle\int{\frac{x~dx}{\sqrt{x^2-a^2}}}$} +$$\int{\frac{x}{\sqrt{x^2-a^2}}}=\sqrt{x^2-a^2}$$ +<<*>>= +)clear all + +--S 2 of 28 +aa:=integrate(x/(sqrt(x^2-a^2)),x) +--R +--R +--R +-------+ +--R | 2 2 2 2 +--R - x\|x - a + x - a +--R (1) ----------------------- +--R +-------+ +--R | 2 2 +--R \|x - a - x +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.211~~~~~$\displaystyle +\int{\frac{x^2~dx}{\sqrt{x^2-a^2}}}$} +$$\int{\frac{x^2}{\sqrt{x^2-a^2}}}= +\frac{x\sqrt{x^2-a^2}}{2}+\frac{a^2}{2}\ln\left(x+\sqrt{x^2-a^2}\right) +$$ +<<*>>= +)clear all + +--S 3 of 28 +aa:=integrate(x^2/sqrt(x^2-a^2),x) +--R +--R +--R (1) +--R +-------+ +-------+ +--R 2 | 2 2 2 2 4 | 2 2 +--R (- 2a x\|x - a + 2a x - a )log(\|x - a - x) +--R + +--R +-------+ +--R 3 2 | 2 2 4 2 2 +--R (- 2x + a x)\|x - a + 2x - 2a x +--R / +--R +-------+ +--R | 2 2 2 2 +--R 4x\|x - a - 4x + 2a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.212~~~~~$\displaystyle +\int{\frac{x^3~dx}{\sqrt{x^2-a^2}}}$} +$$\int{\frac{x^3}{\sqrt{x^2-a^2}}}= +\frac{(x^2-a^2)^{3/2}}{3}+a^2\sqrt{x^2-a^2} +$$ +<<*>>= +)clear all + +--S 4 of 28 +aa:=integrate(x^3/sqrt(x^2-a^2),x) +--R +--R +--R +-------+ +--R 5 2 3 4 | 2 2 6 2 4 4 2 6 +--R (- 4x - 5a x + 6a x)\|x - a + 4x + 3a x - 9a x + 2a +--R (1) ------------------------------------------------------------ +--R +-------+ +--R 2 2 | 2 2 3 2 +--R (12x - 3a )\|x - a - 12x + 9a x +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.213~~~~~$\displaystyle\int{\frac{dx}{x\sqrt{x^2-a^2}}}$} +$$\int{\frac{1}{x\sqrt{x^2-a^2}}}= +\frac{1}{a}\sec^{-1}\left|\frac{x}{a}\right| +$$ +<<*>>= +)clear all + +--S 5 of 28 +aa:=integrate(1/(x*sqrt(x^2-a^2)),x) +--R +--R +--R +-------+ +--R | 2 2 +--R \|x - a - x +--R 2atan(--------------) +--R a +--R (1) --------------------- +--R a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.214~~~~~$\displaystyle +\int{\frac{dx}{x^2\sqrt{x^2-a^2}}}$} +$$\int{\frac{1}{x^2\sqrt{x^2-a^2}}}= +\frac{\sqrt{x^2-a^2}}{a^2x} +$$ +<<*>>= +)clear all + +--S 6 of 28 +aa:=integrate(1/(x^2*sqrt(x^2-a^2)),x) +--R +--R +--R 1 +--R (1) - ---------------- +--R +-------+ +--R | 2 2 2 +--R x\|x - a - x +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.215~~~~~$\displaystyle\int{\frac{dx}{x^3\sqrt{x^2-a^2}}}$} +$$\int{\frac{1}{x^3\sqrt{x^2-a^2}}}= +-\frac{\sqrt{x^2-a^2}}{2a^2x^2}+\frac{1}{2a^3} +\sec^{-1}\left|\frac{x}{a}\right| +$$ +<<*>>= +)clear all + +--S 7 of 28 +aa:=integrate(1/(x^3*sqrt(x^2-a^2)),x) +--R +--R +--R (1) +--R +-------+ +--R +-------+ | 2 2 +--R 3 | 2 2 4 2 2 \|x - a - x +--R (4x \|x - a - 4x + 2a x )atan(--------------) +--R a +--R + +--R +-------+ +--R 2 3 | 2 2 3 3 +--R (- 2a x + a )\|x - a + 2a x - 2a x +--R / +--R +-------+ +--R 3 3 | 2 2 3 4 5 2 +--R 4a x \|x - a - 4a x + 2a x +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.216~~~~~$\displaystyle\int{\sqrt{x^2-a^2}}~dx$} +$$\int{\sqrt{x^2-a^2}}= +\frac{x\sqrt{x^2-a^2}}{2}-\frac{a^2}{2}\ln\left(x+\sqrt{x^2-a^2}\right) +$$ +<<*>>= +)clear all + +--S 8 of 28 +aa:=integrate(sqrt(x^2-a^2),x) +--R +--R +--R (1) +--R +-------+ +-------+ +--R 2 | 2 2 2 2 4 | 2 2 +--R (2a x\|x - a - 2a x + a )log(\|x - a - x) +--R + +--R +-------+ +--R 3 2 | 2 2 4 2 2 +--R (- 2x + a x)\|x - a + 2x - 2a x +--R / +--R +-------+ +--R | 2 2 2 2 +--R 4x\|x - a - 4x + 2a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.217~~~~~$\displaystyle\int{x\sqrt{x^2-a^2}}~dx$} +$$\int{x\sqrt{x^2-a^2}}= +\frac{(x^2-a^2)^{3/2}}{3} +$$ +<<*>>= +)clear all + +--S 9 of 28 +aa:=integrate(x*sqrt(x^2-a^2),x) +--R +--R +--R +-------+ +--R 5 2 3 4 | 2 2 6 2 4 4 2 6 +--R (- 4x + 7a x - 3a x)\|x - a + 4x - 9a x + 6a x - a +--R (1) ----------------------------------------------------------- +--R +-------+ +--R 2 2 | 2 2 3 2 +--R (12x - 3a )\|x - a - 12x + 9a x +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.218~~~~~$\displaystyle +\int{x^2\sqrt{x^2-a^2}}~dx$} +$$\int{x^2\sqrt{x^2-a^2}}= +\frac{x(x^2-a^2)^{3/2}}{4}+\frac{a^2x\sqrt{x^2-a^2}}{8}- +\frac{a^4}{8}\ln\left(x+\sqrt{x^2-a^2}\right) +$$ +<<*>>= +)clear all + +--S 10 of 28 +aa:=integrate(x^2*sqrt(x^2-a^2),x) +--R +--R +--R (1) +--R +-------+ +-------+ +--R 4 3 6 | 2 2 4 4 6 2 8 | 2 2 +--R ((8a x - 4a x)\|x - a - 8a x + 8a x - a )log(\|x - a - x) +--R + +--R +-------+ +--R 7 2 5 4 3 6 | 2 2 8 2 6 4 4 6 2 +--R (- 16x + 24a x - 10a x + a x)\|x - a + 16x - 32a x + 20a x - 4a x +--R / +--R +-------+ +--R 3 2 | 2 2 4 2 2 4 +--R (64x - 32a x)\|x - a - 64x + 64a x - 8a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.219~~~~~$\displaystyle +\int{x^3\sqrt{x^2-a^2}}~dx$} +$$\int{x^3\sqrt{x^2-a^2}}= +\frac{(x^2-a^2)^{5/2}}{5}+\frac{a^2(x^2-a^2)^{3/2}}{3} +$$ +<<*>>= +)clear all + +--S 11 of 28 +aa:=integrate(x^3*sqrt(x^2-a^2),x) +--R +--R +--R (1) +--R +-------+ +--R 9 2 7 4 5 6 3 8 | 2 2 10 2 8 +--R (- 48x + 76a x - 3a x - 35a x + 10a x)\|x - a + 48x - 100a x +--R + +--R 4 6 6 4 8 2 10 +--R 35a x + 40a x - 25a x + 2a +--R / +--R +-------+ +--R 4 2 2 4 | 2 2 5 2 3 4 +--R (240x - 180a x + 15a )\|x - a - 240x + 300a x - 75a x +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.220~~~~~$\displaystyle +\int{\frac{\sqrt{x^2-a^2}}{x}}~dx$} +$$\int{\frac{\sqrt{x^2-a^2}}{x}}= +\sqrt{x^2-a^2}-a\sec^{-1}\left|\frac{x}{a}\right| +$$ +<<*>>= +)clear all + +--S 12 of 28 +aa:=integrate(sqrt(x^2-a^2)/x,x) +--R +--R +--R +-------+ +--R +-------+ | 2 2 +-------+ +--R | 2 2 \|x - a - x | 2 2 2 2 +--R (- 2a\|x - a + 2a x)atan(--------------) - x\|x - a + x - a +--R a +--R (1) ------------------------------------------------------------------- +--R +-------+ +--R | 2 2 +--R \|x - a - x +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.221~~~~~$\displaystyle +\int{\frac{\sqrt{x^2-a^2}}{x^2}}~dx$} +$$\int{\frac{\sqrt{x^2-a^2}}{x^2}}= +-\frac{\sqrt{x^2-a^2}}{x}+\ln\left(x+\sqrt{x^2-a^2}\right) +$$ +<<*>>= +)clear all + +--S 13 of 28 +aa:=integrate(sqrt(x^2-a^2)/x^2,x) +--R +--R +--R +-------+ +-------+ +--R | 2 2 2 | 2 2 2 +--R (- x\|x - a + x )log(\|x - a - x) + a +--R (1) -------------------------------------------- +--R +-------+ +--R | 2 2 2 +--R x\|x - a - x +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.222~~~~~$\displaystyle +\int{\frac{\sqrt{x^2-a^2}}{x^3}}~dx$} +$$\int{\frac{\sqrt{x^2-a^2}}{x^3}}= +-\frac{\sqrt{x^2-a^2}}{2x^2}+\frac{1}{2a} +\sec^{-1}\left|\frac{x}{a}\right| +$$ +<<*>>= +)clear all + +--S 14 of 28 +aa:=integrate(sqrt(x^2-a^2)/x^3,x) +--R +--R +--R (1) +--R +-------+ +--R +-------+ | 2 2 +--R 3 | 2 2 4 2 2 \|x - a - x +--R (4x \|x - a - 4x + 2a x )atan(--------------) +--R a +--R + +--R +-------+ +--R 2 3 | 2 2 3 3 +--R (2a x - a )\|x - a - 2a x + 2a x +--R / +--R +-------+ +--R 3 | 2 2 4 3 2 +--R 4a x \|x - a - 4a x + 2a x +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.223~~~~~$\displaystyle\int{\frac{dx}{(x^2-a^2)^{3/2}}}$} +$$\int{\frac{1}{(x^2-a^2)^{3/2}}}= +-\frac{x}{a^2\sqrt{x^2-a^2}} +$$ +<<*>>= +)clear all + +--S 15 of 28 +aa:=integrate(1/(x^2-a^2)^(3/2),x) +--R +--R +--R 1 +--R (1) - --------------------- +--R +-------+ +--R | 2 2 2 2 +--R x\|x - a - x + a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.224~~~~~$\displaystyle +\int{\frac{x~dx}{(x^2-a^2)^{3/2}}}$} +$$\int{\frac{x}{(x^2-a^2)^{3/2}}}= +\frac{-1}{\sqrt{x^2-a^2}} +$$ +<<*>>= +)clear all + +--S 16 of 28 +aa:=integrate(x/(x^2-a^2)^(3/2),x) +--R +--R +--R +-------+ +--R | 2 2 +--R \|x - a - x +--R (1) --------------------- +--R +-------+ +--R | 2 2 2 2 +--R x\|x - a - x + a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.225~~~~~$\displaystyle +\int{\frac{x^2dx}{(x^2-a^2)^{3/2}}}$} +$$\int{\frac{x^2}{(x^2-a^2)^{3/2}}}= +\frac{-x}{\sqrt{x^2-a^2}}+\ln\left(x+\sqrt{x^2-a^2}\right) +$$ +<<*>>= +)clear all + +--S 17 of 28 +aa:=integrate(x^2/(x^2-a^2)^(3/2),x) +--R +--R +--R +-------+ +-------+ +--R | 2 2 2 2 | 2 2 2 +--R (- x\|x - a + x - a )log(\|x - a - x) - a +--R (1) ------------------------------------------------- +--R +-------+ +--R | 2 2 2 2 +--R x\|x - a - x + a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.226~~~~~$\displaystyle +\int{\frac{x^3dx}{(x^2-a^2)^{3/2}}}$} +$$\int{\frac{x^3}{(x^2-a^2)^{3/2}}}= +\sqrt{x^2-a^2}-\frac{a^2}{\sqrt{x^2-a^2}} +$$ +<<*>>= +)clear all + +--S 18 of 28 +aa:=integrate(x^3/(x^2-a^2)^(3/2),x) +--R +--R +--R +-------+ +--R 3 2 | 2 2 4 2 2 4 +--R (- 2x + 4a x)\|x - a + 2x - 5a x + 2a +--R (1) -------------------------------------------- +--R +-------+ +--R 2 2 | 2 2 3 2 +--R (2x - a )\|x - a - 2x + 2a x +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.227~~~~~$\displaystyle +\int{\frac{dx}{x(x^2-a^2)^{3/2}}}$} +$$\int{\frac{1}{x(x^2-a^2)^{3/2}}}= +\frac{-1}{a^2\sqrt{x^2-a^2}}- +\frac{1}{a^3}\sec^{-1}\left|\frac{x}{a}\right| +$$ +<<*>>= +)clear all + +--S 19 of 28 +aa:=integrate(1/(x*(x^2-a^2)^(3/2)),x) +--R +--R +--R +-------+ +--R +-------+ | 2 2 +-------+ +--R | 2 2 2 2 \|x - a - x | 2 2 +--R (- 2x\|x - a + 2x - 2a )atan(--------------) + a\|x - a - a x +--R a +--R (1) -------------------------------------------------------------------- +--R +-------+ +--R 3 | 2 2 3 2 5 +--R a x\|x - a - a x + a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.228~~~~~$\displaystyle +\int{\frac{dx}{x^2(x^2-a^2)^{3/2}}}$} +$$\int{\frac{1}{x^2(x^2-a^2)^{3/2}}}= +-\frac{\sqrt{x^2-a^2}}{a^4x}-\frac{x}{a^4\sqrt{x^2-a^2}} +$$ +<<*>>= +)clear all + +--S 20 of 28 +aa:=integrate(1/(x^2*(x^2-a^2)^(3/2)),x) +--R +--R +--R 1 +--R (1) - ----------------------------------- +--R +-------+ +--R 3 2 | 2 2 4 2 2 +--R (2x - a x)\|x - a - 2x + 2a x +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.229~~~~~$\displaystyle +\int{\frac{dx}{x^3(x^2-a^2)^{3/2}}}$} +$$\int{\frac{1}{x^3(x^2-a^2)^{3/2}}}= +\frac{1}{2a^2x^2\sqrt{x^2-a^2}}- +\frac{3}{2a^4\sqrt{x^2-a^2}}- +\frac{3}{2a^5}\sec^{-1}\left|\frac{x}{a}\right| +$$ +<<*>>= +)clear all + +--S 21 of 28 +aa:=integrate(1/(x^3*(x^2-a^2)^(3/2)),x) +--R +--R +--R (1) +--R +-------+ +--R +-------+ | 2 2 +--R 5 2 3 | 2 2 6 2 4 4 2 \|x - a - x +--R ((- 24x + 18a x )\|x - a + 24x - 30a x + 6a x )atan(--------------) +--R a +--R + +--R +-------+ +--R 4 3 2 5 | 2 2 5 3 3 5 +--R (12a x - 7a x + a )\|x - a - 12a x + 13a x - 3a x +--R / +--R +-------+ +--R 5 5 7 3 | 2 2 5 6 7 4 9 2 +--R (8a x - 6a x )\|x - a - 8a x + 10a x - 2a x +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.230~~~~~$\displaystyle\int{(x^2-a^2)^{3/2}}~dx$} +$$\int{(x^2-a^2)^{3/2}}= +\frac{x(x^2-a^2)^{3/2}}{4}-\frac{3a^2x\sqrt{x^2-a^2}}{8}+ +\frac{3}{8}a^4\ln\left(x+\sqrt{x^2-a^2}\right) +$$ +<<*>>= +)clear all + +--S 22 of 28 +aa:=integrate((x^2-a^2)^(3/2),x) +--R +--R +--R (1) +--R +-------+ +-------+ +--R 4 3 6 | 2 2 4 4 6 2 8 | 2 2 +--R ((- 24a x + 12a x)\|x - a + 24a x - 24a x + 3a )log(\|x - a - x) +--R + +--R +-------+ +--R 7 2 5 4 3 6 | 2 2 8 2 6 4 4 +--R (- 16x + 56a x - 42a x + 5a x)\|x - a + 16x - 64a x + 68a x +--R + +--R 6 2 +--R - 20a x +--R / +--R +-------+ +--R 3 2 | 2 2 4 2 2 4 +--R (64x - 32a x)\|x - a - 64x + 64a x - 8a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.231~~~~~$\displaystyle\int{x(x^2-a^2)^{3/2}}~dx$} +$$\int{x(x^2-a^2)^{3/2}}=\frac{(x^2-a^2)^{5/2}}{5}$$ +<<*>>= +)clear all + +--S 23 of 28 +aa:=integrate(x*(x^2-a^2)^(3/2),x) +--R +--R +--R (1) +--R +-------+ +--R 9 2 7 4 5 6 3 8 | 2 2 10 2 8 +--R (- 16x + 52a x - 61a x + 30a x - 5a x)\|x - a + 16x - 60a x +--R + +--R 4 6 6 4 8 2 10 +--R 85a x - 55a x + 15a x - a +--R / +--R +-------+ +--R 4 2 2 4 | 2 2 5 2 3 4 +--R (80x - 60a x + 5a )\|x - a - 80x + 100a x - 25a x +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.232~~~~~$\displaystyle\int{x^2(x^2-a^2)^{3/2}}~dx$} +$$\int{x^2(x^2-a^2)^{3/2}}= +\frac{x(x^2-a^2)^{5/2}}{6}+\frac{a^2x(x^2-a^2)^{3/2}}{24}- +\frac{a^4x\sqrt{x^2-a^2}}{16}+ +\frac{a^6}{16}\ln\left(x+\sqrt{x^2-a^2}\right) +$$ +<<*>>= +)clear all + +--S 24 of 28 +aa:=integrate(x^2*(x^2-a^2)^(3/2),x) +--R +--R +--R (1) +--R +-------+ +--R 6 5 8 3 10 | 2 2 6 6 8 4 10 2 +--R (- 96a x + 96a x - 18a x)\|x - a + 96a x - 144a x + 54a x +--R + +--R 12 +--R - 3a +--R * +--R +-------+ +--R | 2 2 +--R log(\|x - a - x) +--R + +--R +-------+ +--R 11 2 9 4 7 6 5 8 3 10 | 2 2 +--R (- 256x + 832a x - 912a x + 404a x - 68a x + 3a x)\|x - a +--R + +--R 12 2 10 4 8 6 6 8 4 10 2 +--R 256x - 960a x + 1296a x - 772a x + 198a x - 18a x +--R / +--R +-------+ +--R 5 2 3 4 | 2 2 6 2 4 4 2 6 +--R (1536x - 1536a x + 288a x)\|x - a - 1536x + 2304a x - 864a x + 48a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.233~~~~~$\displaystyle\int{x^3(x^2-a^2)^{3/2}}~dx$} +$$\int{x^3(x^2-a^2)^{3/2}}= +\frac{(x^2-a^2)^{7/2}}{7}+\frac{a^2(x^2-a^2)^{5/2}}{5} +$$ +<<*>>= +)clear all + +--S 25 of 28 +aa:=integrate(x^3*(x^2-a^2)^(3/2),x) +--R +--R +--R (1) +--R 13 2 11 4 9 6 7 8 5 10 3 +--R - 320x + 1072a x - 1240a x + 467a x + 112a x - 105a x +--R + +--R 12 +--R 14a x +--R * +--R +-------+ +--R | 2 2 +--R \|x - a +--R + +--R 14 2 12 4 10 6 8 8 6 10 4 12 2 +--R 320x - 1232a x + 1736a x - 973a x + 21a x + 175a x - 49a x +--R + +--R 14 +--R 2a +--R / +--R +-------+ +--R 6 2 4 4 2 6 | 2 2 7 2 5 +--R (2240x - 2800a x + 840a x - 35a )\|x - a - 2240x + 3920a x +--R + +--R 4 3 6 +--R - 1960a x + 245a x +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.234~~~~~$\displaystyle +\int{\frac{(x^2-a^2)^{3/2}}{x}}~dx$} +$$\int{\frac{(x^2-a^2)^{3/2}}{x}}= +\frac{(x^2-a^2)^{3/2}}{3}-a^2\sqrt{x^2-a^2}+ +a^3\sec^{-1}\left|\frac{x}{a}\right| +$$ +<<*>>= +)clear all + +--S 26 of 28 +aa:=integrate((x^2-a^2)^(3/2)/x,x) +--R +--R +--R (1) +--R +-------+ +--R +-------+ | 2 2 +--R 3 2 5 | 2 2 3 3 5 \|x - a - x +--R ((24a x - 6a )\|x - a - 24a x + 18a x)atan(--------------) +--R a +--R + +--R +-------+ +--R 5 2 3 4 | 2 2 6 2 4 4 2 6 +--R (- 4x + 19a x - 12a x)\|x - a + 4x - 21a x + 21a x - 4a +--R / +--R +-------+ +--R 2 2 | 2 2 3 2 +--R (12x - 3a )\|x - a - 12x + 9a x +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.235~~~~~$\displaystyle +\int{\frac{(x^2-a^2)^{3/2}}{x^2}}~dx$} +$$\int{\frac{(x^2-a^2)^{3/2}}{x^2}}= +-\frac{(x^2-a^2)^{3/2}}{x}+\frac{3x\sqrt{x^2-a^2}}{2}- +\frac{3}{2}a^2\ln\left(x+\sqrt{x^2-a^2}\right) +$$ +<<*>>= +)clear all + +--S 27 of 28 +aa:=integrate((x^2-a^2)^{3/2}/x^2,x) +--R +--R +--R (1) +--R +-------+ +-------+ +--R 2 3 4 | 2 2 2 4 4 2 | 2 2 +--R ((12a x - 3a x)\|x - a - 12a x + 9a x )log(\|x - a - x) +--R + +--R +-------+ +--R 5 2 3 4 | 2 2 6 2 4 4 2 6 +--R (- 4x + 3a x + 4a x)\|x - a + 4x - 5a x - 3a x + 2a +--R / +--R +-------+ +--R 3 2 | 2 2 4 2 2 +--R (8x - 2a x)\|x - a - 8x + 6a x +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.236~~~~~$\displaystyle +\int{\frac{(x^2-a^2)^{3/2}}{x^3}}~dx$} +$$\int{\frac{(x^2-a^2)^{3/2}}{x^3}}= +-\frac{(x^2-a^2)^{3/2}}{2x^2}+\frac{3}{2}\sqrt{x^2-a^2}- +\frac{3}{2}a\sec^{-1}\left|\frac{x}{a}\right| +$$ +<<*>>= +)clear all + +--S 28 of 28 +aa:=integrate((x^2-a^2)^(3/2)/x^3,x) +--R +--R +--R (1) +--R +-------+ +--R +-------+ | 2 2 +--R 4 3 2 | 2 2 5 3 3 \|x - a - x +--R ((- 24a x + 6a x )\|x - a + 24a x - 18a x )atan(--------------) +--R a +--R + +--R +-------+ +--R 5 2 3 4 | 2 2 6 2 4 4 2 6 +--R (- 8x + 2a x + 3a x)\|x - a + 8x - 6a x - 3a x + a +--R / +--R +-------+ +--R 4 2 2 | 2 2 5 2 3 +--R (8x - 2a x )\|x - a - 8x + 6a x +--R Type: Union(Expression Integer,...) +--E + +)spool +)lisp (bye) +@ + +\eject +\begin{thebibliography}{99} +\bibitem{1} Spiegel, Murray R. +{\sl Mathematical Handbook of Formulas and Tables}\\ +Schaum's Outline Series McGraw-Hill 1968 pp68-69 +\end{thebibliography} +\end{document} diff --git a/src/input/schaum11.input.pamphlet b/src/input/schaum11.input.pamphlet new file mode 100644 index 0000000..6cd59a6 --- /dev/null +++ b/src/input/schaum11.input.pamphlet @@ -0,0 +1,783 @@ +\documentclass{article} +\usepackage{axiom} +\begin{document} +\title{\$SPAD/input schaum11.input} +\author{Timothy Daly} +\maketitle +\eject +\tableofcontents +\eject +\section{\cite{1}:14.237~~~~~$\displaystyle\int{\frac{dx}{\sqrt{a^2-x^2}}}$} +$$\int{\frac{1}{\sqrt{a^2-x^2}}}=\ln\left(x+\sqrt{a^2-x^2}\right)$$ +<<*>>= +)spool schaum11.output +)set message test on +)set message auto off +)clear all + +--S 1 of 28 +aa:=integrate(1/(sqrt(a^2-x^2)),x) +--R +--R +--R +---------+ +--R | 2 2 +--R \|- x + a - a +--R (1) - 2atan(----------------) +--R x +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.238~~~~~$\displaystyle\int{\frac{x~dx}{\sqrt{a^2-x^2}}}$} +$$\int{\frac{x}{\sqrt{a^2-x^2}}}=\sqrt{a^2-x^2}$$ +<<*>>= +)clear all + +--S 2 of 28 +aa:=integrate(x/(sqrt(a^2-x^2)),x) +--R +--R +--R 2 +--R x +--R (1) ---------------- +--R +---------+ +--R | 2 2 +--R \|- x + a - a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.239~~~~~$\displaystyle +\int{\frac{x^2~dx}{\sqrt{a^2-x^2}}}$} +$$\int{\frac{x^2}{\sqrt{a^2-x^2}}}= +\frac{x\sqrt{a^2-x^2}}{2}+\frac{a^2}{2}\ln\left(x+\sqrt{a^2-x^2}\right) +$$ +<<*>>= +)clear all + +--S 3 of 28 +aa:=integrate(x^2/sqrt(a^2-x^2),x) +--R +--R +--R (1) +--R +---------+ +--R +---------+ | 2 2 +--R 3 | 2 2 2 2 4 \|- x + a - a +--R (- 4a \|- x + a - 2a x + 4a )atan(----------------) +--R x +--R + +--R +---------+ +--R 3 2 | 2 2 3 3 +--R (- x + 2a x)\|- x + a + 2a x - 2a x +--R / +--R +---------+ +--R | 2 2 2 2 +--R 4a\|- x + a + 2x - 4a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.240~~~~~$\displaystyle +\int{\frac{x^3~dx}{\sqrt{a^2-x^2}}}$} +$$\int{\frac{x^3}{\sqrt{a^2-x^2}}}= +\frac{(a^2-x^2)^{3/2}}{3}+a^2\sqrt{a^2-x^2} +$$ +<<*>>= +)clear all + +--S 4 of 28 +aa:=integrate(x^3/sqrt(a^2-x^2),x) +--R +--R +--R +---------+ +--R 4 | 2 2 6 2 4 +--R 3a x \|- x + a + x - 3a x +--R (1) --------------------------------------- +--R +---------+ +--R 2 2 | 2 2 2 3 +--R (3x - 12a )\|- x + a - 9a x + 12a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.241~~~~~$\displaystyle\int{\frac{dx}{x\sqrt{a^2-x^2}}}$} +$$\int{\frac{1}{x\sqrt{a^2-x^2}}}= +\frac{1}{a}\sec^{-1}\left|\frac{x}{a}\right| +$$ +<<*>>= +)clear all + +--S 5 of 28 +aa:=integrate(1/(x*sqrt(a^2-x^2)),x) +--R +--R +--R +---------+ +--R | 2 2 +--R \|- x + a - a +--R log(----------------) +--R x +--R (1) --------------------- +--R a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.242~~~~~$\displaystyle +\int{\frac{dx}{x^2\sqrt{a^2-x^2}}}$} +$$\int{\frac{1}{x^2\sqrt{a^2-x^2}}}= +\frac{\sqrt{a^2-x^2}}{a^2x} +$$ +<<*>>= +)clear all + +--S 6 of 28 +aa:=integrate(1/(x^2*sqrt(a^2-x^2)),x) +--R +--R +--R +---------+ +--R | 2 2 2 2 +--R a\|- x + a + x - a +--R (1) ----------------------- +--R +---------+ +--R 2 | 2 2 3 +--R a x\|- x + a - a x +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.243~~~~~$\displaystyle\int{\frac{dx}{x^3\sqrt{a^2-x^2}}}$} +$$\int{\frac{1}{x^3\sqrt{a^2-x^2}}}= +-\frac{\sqrt{a^2-x^2}}{2a^2x^2}+\frac{1}{2a^3} +\sec^{-1}\left|\frac{x}{a}\right| +$$ +<<*>>= +)clear all + +--S 7 of 28 +aa:=integrate(1/(x^3*sqrt(a^2-x^2)),x) +--R +--R +--R (1) +--R +---------+ +--R +---------+ | 2 2 +--R 2 | 2 2 4 2 2 \|- x + a - a +--R (2a x \|- x + a + x - 2a x )log(----------------) +--R x +--R + +--R +---------+ +--R 2 3 | 2 2 2 2 4 +--R (- a x + 2a )\|- x + a + 2a x - 2a +--R / +--R +---------+ +--R 4 2 | 2 2 3 4 5 2 +--R 4a x \|- x + a + 2a x - 4a x +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.244~~~~~$\displaystyle\int{\sqrt{a^2-x^2}}~dx$} +$$\int{\sqrt{a^2-x^2}}= +\frac{x\sqrt{a^2-x^2}}{2}-\frac{a^2}{2}\ln\left(x+\sqrt{a^2-x^2}\right) +$$ +<<*>>= +)clear all + +--S 8 of 28 +aa:=integrate(sqrt(a^2-x^2),x) +--R +--R +--R (1) +--R +---------+ +--R +---------+ | 2 2 +--R 3 | 2 2 2 2 4 \|- x + a - a +--R (- 4a \|- x + a - 2a x + 4a )atan(----------------) +--R x +--R + +--R +---------+ +--R 3 2 | 2 2 3 3 +--R (x - 2a x)\|- x + a - 2a x + 2a x +--R / +--R +---------+ +--R | 2 2 2 2 +--R 4a\|- x + a + 2x - 4a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.245~~~~~$\displaystyle\int{x\sqrt{a^2-x^2}}~dx$} +$$\int{x\sqrt{a^2-x^2}}= +\frac{(a^2-x^2)^{3/2}}{3} +$$ +<<*>>= +)clear all + +--S 9 of 28 +aa:=integrate(x*sqrt(a^2-x^2),x) +--R +--R +--R +---------+ +--R 4 3 2 | 2 2 6 2 4 4 2 +--R (- 3a x + 6a x )\|- x + a - x + 6a x - 6a x +--R (1) -------------------------------------------------- +--R +---------+ +--R 2 2 | 2 2 2 3 +--R (3x - 12a )\|- x + a - 9a x + 12a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.246~~~~~$\displaystyle +\int{x^2\sqrt{a^2-x^2}}~dx$} +$$\int{x^2\sqrt{a^2-x^2}}= +\frac{x(a^2-x^2)^{3/2}}{4}+\frac{a^2x\sqrt{a^2-x^2}}{8}- +\frac{a^4}{8}\ln\left(x+\sqrt{a^2-x^2}\right) +$$ +<<*>>= +)clear all + +--S 10 of 28 +aa:=integrate(x^2*sqrt(a^2-x^2),x) +--R +--R +--R (1) +--R +---------+ +--R 5 2 7 | 2 2 4 4 6 2 8 +--R ((- 8a x + 16a )\|- x + a - 2a x + 16a x - 16a ) +--R * +--R +---------+ +--R | 2 2 +--R \|- x + a - a +--R atan(----------------) +--R x +--R + +--R +---------+ +--R 7 2 5 4 3 6 | 2 2 7 3 5 5 3 7 +--R (2x - 17a x + 24a x - 8a x)\|- x + a - 8a x + 28a x - 28a x + 8a x +--R / +--R +---------+ +--R 2 3 | 2 2 4 2 2 4 +--R (32a x - 64a )\|- x + a + 8x - 64a x + 64a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.247~~~~~$\displaystyle +\int{x^3\sqrt{a^2-x^2}}~dx$} +$$\int{x^3\sqrt{a^2-x^2}}= +\frac{(a^2-x^2)^{5/2}}{5}+\frac{a^2(a^2-x^2)^{3/2}}{3} +$$ +<<*>>= +)clear all + +--S 11 of 28 +aa:=integrate(x^3*sqrt(a^2-x^2),x) +--R +--R +--R (1) +--R +---------+ +--R 8 3 6 5 4 | 2 2 10 2 8 4 6 6 4 +--R (- 15a x + 65a x - 60a x )\|- x + a - 3x + 40a x - 95a x + 60a x +--R -------------------------------------------------------------------------- +--R +---------+ +--R 4 2 2 4 | 2 2 4 3 2 5 +--R (15x - 180a x + 240a )\|- x + a - 75a x + 300a x - 240a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.248~~~~~$\displaystyle +\int{\frac{\sqrt{a^2-x^2}}{x}}~dx$} +$$\int{\frac{\sqrt{a^2-x^2}}{x}}= +\sqrt{a^2-x^2}-a\sec^{-1}\left|\frac{x}{a}\right| +$$ +<<*>>= +)clear all + +--S 12 of 28 +aa:=integrate(sqrt(a^2-x^2)/x,x) +--R +--R +--R +---------+ +--R +---------+ | 2 2 +--R | 2 2 2 \|- x + a - a 2 +--R (a\|- x + a - a )log(----------------) - x +--R x +--R (1) ---------------------------------------------- +--R +---------+ +--R | 2 2 +--R \|- x + a - a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.249~~~~~$\displaystyle +\int{\frac{\sqrt{a^2-x^2}}{x^2}}~dx$} +$$\int{\frac{\sqrt{a^2-x^2}}{x^2}}= +-\frac{\sqrt{a^2-x^2}}{x}+\ln\left(x+\sqrt{a^2-x^2}\right) +$$ +<<*>>= +)clear all + +--S 13 of 28 +aa:=integrate(sqrt(a^2-x^2)/x^2,x) +--R +--R +--R (1) +--R +---------+ +--R +---------+ | 2 2 +---------+ +--R | 2 2 \|- x + a - a | 2 2 2 2 +--R (2x\|- x + a - 2a x)atan(----------------) + a\|- x + a + x - a +--R x +--R ----------------------------------------------------------------------- +--R +---------+ +--R | 2 2 +--R x\|- x + a - a x +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.250~~~~~$\displaystyle +\int{\frac{\sqrt{a^2-x^2}}{x^3}}~dx$} +$$\int{\frac{\sqrt{a^2-x^2}}{x^3}}= +-\frac{\sqrt{a^2-x^2}}{2x^2}+\frac{1}{2a} +\sec^{-1}\left|\frac{x}{a}\right| +$$ +<<*>>= +)clear all + +--S 14 of 28 +aa:=integrate(sqrt(a^2-x^2)/x^3,x) +--R +--R +--R (1) +--R +---------+ +--R +---------+ | 2 2 +--R 2 | 2 2 4 2 2 \|- x + a - a +--R (- 2a x \|- x + a - x + 2a x )log(----------------) +--R x +--R + +--R +---------+ +--R 2 3 | 2 2 2 2 4 +--R (- a x + 2a )\|- x + a + 2a x - 2a +--R / +--R +---------+ +--R 2 2 | 2 2 4 3 2 +--R 4a x \|- x + a + 2a x - 4a x +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.251~~~~~$\displaystyle\int{\frac{dx}{(a^2-x^2)^{3/2}}}$} +$$\int{\frac{1}{(a^2-x^2)^{3/2}}}= +-\frac{x}{a^2\sqrt{a^2-x^2}} +$$ +<<*>>= +)clear all + +--S 15 of 28 +aa:=integrate(1/(a^2-x^2)^(3/2),x) +--R +--R +--R +---------+ +--R | 2 2 +--R - x\|- x + a + a x +--R (1) -------------------------- +--R +---------+ +--R 3 | 2 2 2 2 4 +--R a \|- x + a + a x - a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.252~~~~~$\displaystyle +\int{\frac{x~dx}{(a^2-x^2)^{3/2}}}$} +$$\int{\frac{x}{(a^2-x^2)^{3/2}}}= +\frac{-1}{\sqrt{a^2-x^2}} +$$ +<<*>>= +)clear all + +--S 16 of 28 +aa:=integrate(x/(a^2-x^2)^(3/2),x) +--R +--R +--R 2 +--R x +--R (1) -------------------------- +--R +---------+ +--R 2 | 2 2 2 3 +--R a \|- x + a + a x - a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.253~~~~~$\displaystyle +\int{\frac{x^2dx}{(a^2-x^2)^{3/2}}}$} +$$\int{\frac{x^2}{(a^2-x^2)^{3/2}}}= +\frac{-x}{\sqrt{a^2-x^2}}+\ln\left(x+\sqrt{a^2-x^2}\right) +$$ +<<*>>= +)clear all + +--S 17 of 28 +aa:=integrate(x^2/(a^2-x^2)^(3/2),x) +--R +--R +--R (1) +--R +---------+ +--R +---------+ | 2 2 +---------+ +--R | 2 2 2 2 \|- x + a - a | 2 2 +--R (2a\|- x + a + 2x - 2a )atan(----------------) - x\|- x + a + a x +--R x +--R ------------------------------------------------------------------------ +--R +---------+ +--R | 2 2 2 2 +--R a\|- x + a + x - a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.254~~~~~$\displaystyle +\int{\frac{x^3dx}{(a^2-x^2)^{3/2}}}$} +$$\int{\frac{x^3}{(a^2-x^2)^{3/2}}}= +\sqrt{a^2-x^2}-\frac{a^2}{\sqrt{a^2-x^2}} +$$ +<<*>>= +)clear all + +--S 18 of 28 +aa:=integrate(x^3/(a^2-x^2)^(3/2),x) +--R +--R +--R 4 +--R x +--R (1) - ------------------------------------ +--R +---------+ +--R 2 2 | 2 2 2 3 +--R (x - 2a )\|- x + a - 2a x + 2a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.255~~~~~$\displaystyle +\int{\frac{dx}{x(a^2-x^2)^{3/2}}}$} +$$\int{\frac{1}{x(a^2-x^2)^{3/2}}}= +\frac{-1}{a^2\sqrt{a^2-x^2}}- +\frac{1}{a^3}\sec^{-1}\left|\frac{x}{a}\right| +$$ +<<*>>= +)clear all + +--S 19 of 28 +aa:=integrate(1/(x*(a^2-x^2)^(3/2)),x) +--R +--R +--R +---------+ +--R +---------+ | 2 2 +--R | 2 2 2 2 \|- x + a - a 2 +--R (a\|- x + a + x - a )log(----------------) + x +--R x +--R (1) --------------------------------------------------- +--R +---------+ +--R 4 | 2 2 3 2 5 +--R a \|- x + a + a x - a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.256~~~~~$\displaystyle +\int{\frac{dx}{x^2(a^2-x^2)^{3/2}}}$} +$$\int{\frac{1}{x^2(a^2-x^2)^{3/2}}}= +-\frac{\sqrt{a^2-x^2}}{a^4x}-\frac{x}{a^4\sqrt{a^2-x^2}} +$$ +<<*>>= +)clear all + +--S 20 of 28 +aa:=integrate(1/(x^2*(a^2-x^2)^(3/2)),x) +--R +--R +--R +---------+ +--R 2 3 | 2 2 4 2 2 4 +--R (4a x - 2a )\|- x + a + 2x - 5a x + 2a +--R (1) --------------------------------------------- +--R +---------+ +--R 4 3 6 | 2 2 5 3 7 +--R (a x - 2a x)\|- x + a - 2a x + 2a x +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.257~~~~~$\displaystyle +\int{\frac{dx}{x^3(a^2-x^2)^{3/2}}}$} +$$\int{\frac{1}{x^3(a^2-x^2)^{3/2}}}= +\frac{1}{2a^2x^2\sqrt{a^2-x^2}}- +\frac{3}{2a^4\sqrt{a^2-x^2}}- +\frac{3}{2a^5}\sec^{-1}\left|\frac{x}{a}\right| +$$ +<<*>>= +)clear all + +--S 21 of 28 +aa:=integrate(1/(x^3*(a^2-x^2)^(3/2)),x) +--R +--R +--R (1) +--R +---------+ +--R 4 3 2 | 2 2 6 2 4 4 2 +--R ((9a x - 12a x )\|- x + a + 3x - 15a x + 12a x ) +--R * +--R +---------+ +--R | 2 2 +--R \|- x + a - a +--R log(----------------) +--R x +--R + +--R +---------+ +--R 4 3 2 5 | 2 2 6 2 4 4 2 6 +--R (3a x + 5a x - 4a )\|- x + a + 2x - a x - 7a x + 4a +--R / +--R +---------+ +--R 6 4 8 2 | 2 2 5 6 7 4 9 2 +--R (6a x - 8a x )\|- x + a + 2a x - 10a x + 8a x +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.258~~~~~$\displaystyle\int{(a^2-x^2)^{3/2}}~dx$} +$$\int{(a^2-x^2)^{3/2}}= +\frac{x(a^2-x^2)^{3/2}}{4}-\frac{3a^2x\sqrt{a^2-x^2}}{8}+ +\frac{3}{8}a^4\ln\left(x+\sqrt{a^2-x^2}\right) +$$ +<<*>>= +)clear all + +--S 22 of 28 +aa:=integrate((a^2-x^2)^(3/2),x) +--R +--R +--R (1) +--R +---------+ +--R 5 2 7 | 2 2 4 4 6 2 8 +--R ((- 24a x + 48a )\|- x + a - 6a x + 48a x - 48a ) +--R * +--R +---------+ +--R | 2 2 +--R \|- x + a - a +--R atan(----------------) +--R x +--R + +--R +---------+ +--R 7 2 5 4 3 6 | 2 2 7 3 5 5 3 +--R (- 2x + 21a x - 56a x + 40a x)\|- x + a + 8a x - 44a x + 76a x +--R + +--R 7 +--R - 40a x +--R / +--R +---------+ +--R 2 3 | 2 2 4 2 2 4 +--R (32a x - 64a )\|- x + a + 8x - 64a x + 64a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.259~~~~~$\displaystyle\int{x(a^2-x^2)^{3/2}}~dx$} +$$\int{x(a^2-x^2)^{3/2}}=\frac{(a^2-x^2)^{5/2}}{5}$$ +<<*>>= +)clear all + +--S 23 of 28 +aa:=integrate(x*(a^2-x^2)^(3/2),x) +--R +--R +--R (1) +--R +---------+ +--R 8 3 6 5 4 7 2 | 2 2 10 2 8 4 6 +--R (5a x - 30a x + 60a x - 40a x )\|- x + a + x - 15a x + 55a x +--R + +--R 6 4 8 2 +--R - 80a x + 40a x +--R / +--R +---------+ +--R 4 2 2 4 | 2 2 4 3 2 5 +--R (5x - 60a x + 80a )\|- x + a - 25a x + 100a x - 80a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.260~~~~~$\displaystyle\int{x^2(a^2-x^2)^{3/2}}~dx$} +$$\int{x^2(a^2-x^2)^{3/2}}= +\frac{x(a^2-x^2)^{5/2}}{6}+\frac{a^2x(a^2-x^2)^{3/2}}{24}- +\frac{a^4x\sqrt{a^2-x^2}}{16}+ +\frac{a^6}{16}\ln\left(x+\sqrt{a^2-x^2}\right) +$$ +<<*>>= +)clear all + +--S 24 of 28 +aa:=integrate(x^2*(a^2-x^2)^(3/2),x) +--R +--R +--R (1) +--R +---------+ +--R 7 4 9 2 11 | 2 2 6 6 8 4 +--R (- 36a x + 192a x - 192a )\|- x + a - 6a x + 108a x +--R + +--R 10 2 12 +--R - 288a x + 192a +--R * +--R +---------+ +--R | 2 2 +--R \|- x + a - a +--R atan(----------------) +--R x +--R + +--R +---------+ +--R 11 2 9 4 7 6 5 8 3 10 | 2 2 +--R (- 8x + 158a x - 639a x + 982a x - 592a x + 96a x)\|- x + a +--R + +--R 11 3 9 5 7 7 5 9 3 11 +--R 48a x - 388a x + 1062a x - 1266a x + 640a x - 96a x +--R / +--R +---------+ +--R 4 3 2 5 | 2 2 6 2 4 4 2 +--R (288a x - 1536a x + 1536a )\|- x + a + 48x - 864a x + 2304a x +--R + +--R 6 +--R - 1536a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.261~~~~~$\displaystyle\int{x^3(a^2-x^2)^{3/2}}~dx$} +$$\int{x^3(a^2-x^2)^{3/2}}= +\frac{(a^2-x^2)^{7/2}}{7}+\frac{a^2(a^2-x^2)^{5/2}}{5} +$$ +<<*>>= +)clear all + +--S 25 of 28 +aa:=integrate(x^3*(a^2-x^2)^(3/2),x) +--R +--R +--R (1) +--R +---------+ +--R 12 3 10 5 8 7 6 9 4 | 2 2 14 +--R (35a x - 336a x + 1015a x - 1260a x + 560a x )\|- x + a + 5x +--R + +--R 2 12 4 10 6 8 8 6 10 4 +--R - 133a x + 721a x - 1575a x + 1540a x - 560a x +--R / +--R +---------+ +--R 6 2 4 4 2 6 | 2 2 6 3 4 +--R (35x - 840a x + 2800a x - 2240a )\|- x + a - 245a x + 1960a x +--R + +--R 5 2 7 +--R - 3920a x + 2240a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.262~~~~~$\displaystyle +\int{\frac{(a^2-x^2)^{3/2}}{x}}~dx$} +$$\int{\frac{(a^2-x^2)^{3/2}}{x}}= +\frac{(a^2-x^2)^{3/2}}{3}-a^2\sqrt{a^2-x^2}+ +a^3\sec^{-1}\left|\frac{x}{a}\right| +$$ +<<*>>= +)clear all + +--S 26 of 28 +aa:=integrate((a^2-x^2)^(3/2)/x,x) +--R +--R +--R (1) +--R +---------+ +--R +---------+ | 2 2 +--R 3 2 5 | 2 2 4 2 6 \|- x + a - a +--R ((3a x - 12a )\|- x + a - 9a x + 12a )log(----------------) +--R x +--R + +--R +---------+ +--R 4 3 2 | 2 2 6 2 4 4 2 +--R (3a x - 12a x )\|- x + a + x - 9a x + 12a x +--R / +--R +---------+ +--R 2 2 | 2 2 2 3 +--R (3x - 12a )\|- x + a - 9a x + 12a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.263~~~~~$\displaystyle +\int{\frac{(a^2-x^2)^{3/2}}{x^2}}~dx$} +$$\int{\frac{(a^2-x^2)^{3/2}}{x^2}}= +-\frac{(a^2-x^2)^{3/2}}{x}+\frac{3x\sqrt{a^2-x^2}}{2}- +\frac{3}{2}a^2\ln\left(x+\sqrt{a^2-x^2}\right) +$$ +<<*>>= +)clear all + +--S 27 of 28 +aa:=integrate((a^2-x^2)^{3/2}/x^2,x) +--R +--R +--R (1) +--R +---------+ +--R +---------+ | 2 2 +--R 2 3 4 | 2 2 3 3 5 \|- x + a - a +--R ((6a x - 24a x)\|- x + a - 18a x + 24a x)atan(----------------) +--R x +--R + +--R +---------+ +--R 4 3 2 5 | 2 2 6 2 4 4 2 6 +--R (3a x + 2a x - 8a )\|- x + a + x - 3a x - 6a x + 8a +--R / +--R +---------+ +--R 3 2 | 2 2 3 3 +--R (2x - 8a x)\|- x + a - 6a x + 8a x +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.264~~~~~$\displaystyle +\int{\frac{(a^2-x^2)^{3/2}}{x^3}}~dx$} +$$\int{\frac{(a^2-x^2)^{3/2}}{x^3}}= +-\frac{(a^2-x^2)^{3/2}}{2x^2}+\frac{3}{2}\sqrt{a^2-x^2}- +\frac{3}{2}a\sec^{-1}\left|\frac{x}{a}\right| +$$ +<<*>>= +)clear all + +--S 28 of 28 +aa:=integrate((a^2-x^2)^(3/2)/x^3,x) +--R +--R +--R (1) +--R +---------+ +--R +---------+ | 2 2 +--R 4 3 2 | 2 2 2 4 4 2 \|- x + a - a +--R ((- 3a x + 12a x )\|- x + a + 9a x - 12a x )log(----------------) +--R x +--R + +--R +---------+ +--R 4 3 2 5 | 2 2 6 2 4 4 2 6 +--R (4a x + 3a x - 4a )\|- x + a + 2x - 3a x - 5a x + 4a +--R / +--R +---------+ +--R 4 2 2 | 2 2 4 3 2 +--R (2x - 8a x )\|- x + a - 6a x + 8a x +--R Type: Union(Expression Integer,...) +--E + +)spool +)lisp (bye) +@ + +\eject +\begin{thebibliography}{99} +\bibitem{1} Spiegel, Murray R. +{\sl Mathematical Handbook of Formulas and Tables}\\ +Schaum's Outline Series McGraw-Hill 1968 pp68-69 +\end{thebibliography} +\end{document} diff --git a/src/input/schaum2.input.pamphlet b/src/input/schaum2.input.pamphlet index cb8e6db..ba16925 100644 --- a/src/input/schaum2.input.pamphlet +++ b/src/input/schaum2.input.pamphlet @@ -8,7 +8,7 @@ \tableofcontents \eject \section{\cite{1}:14.84~~~~~$\displaystyle\int{\frac{dx}{\sqrt{ax+b}}}$} -$$\int{\frac{dx}{\sqrt{ax+b}}}=\frac{2\sqrt{ax+b}}{a}$$ +$$\int{\frac{1}{\sqrt{ax+b}}}=\frac{2\sqrt{ax+b}}{a}$$ <<*>>= )spool schaum2.output )set message test on @@ -49,7 +49,7 @@ cc:=aa-bb @ \section{\cite{1}:14.85~~~~~$\displaystyle\int{\frac{x~dx}{\sqrt{ax+b}}}$} -$$\int{\frac{x~dx}{\sqrt{ax+b}}}=\frac{2(ax-2b)}{3a^2}\sqrt{ax+b}$$ +$$\int{\frac{x}{\sqrt{ax+b}}}=\frac{2(ax-2b)}{3a^2}\sqrt{ax+b}$$ <<*>>= )clear all @@ -89,7 +89,7 @@ cc:=aa-bb @ \section{\cite{1}:14.86~~~~~$\displaystyle\int{\frac{x^2~dx}{\sqrt{ax+b}}}$} -$$\int{\frac{x~dx}{\sqrt{ax+b}}}= +$$\int{\frac{x}{\sqrt{ax+b}}}= \frac{2(3a^2x^2-4abx+8b^2)}{15a^2}\sqrt{ax+b}$$ <<*>>= )clear all @@ -130,7 +130,7 @@ cc:=aa-bb @ \section{\cite{1}:14.87~~~~~$\displaystyle\int{\frac{dx}{x\sqrt{ax+b}}}$} -$$\int{\frac{dx}{x\sqrt{ax+b}}}= +$$\int{\frac{1}{x\sqrt{ax+b}}}= \left\{ \begin{array}{l} \displaystyle @@ -380,9 +380,9 @@ Thus the original equation and Spiegel's derivative of the integral are equal. So we can conclude that both second answers are correct although they differ by a constant of integration. - \section{\cite{1}:14.88~~~~~$\displaystyle\int{\frac{dx}{x^2\sqrt{ax+b}}}$} -$$\int{\frac{dx}{x^2\sqrt{ax+b}}}= --\frac{\sqrt{ax+b}}{bx}-\frac{a}{2b}~\int{\frac{dx}{x\sqrt{ax+b}}}$$ +\section{\cite{1}:14.88~~~~~$\displaystyle\int{\frac{dx}{x^2\sqrt{ax+b}}}$} +$$\int{\frac{1}{x^2\sqrt{ax+b}}}= +-\frac{\sqrt{ax+b}}{bx}-\frac{a}{2b}~\int{\frac{1}{x\sqrt{ax+b}}}$$ <<*>>= )clear all @@ -565,7 +565,7 @@ cc22:=bb2-aa.2 @ \section{\cite{1}:14.89~~~~~$\displaystyle\int{\sqrt{ax+b}~dx}$} -$$\int{\sqrt{ax+b}~dx}= +$$\int{\sqrt{ax+b}}= \frac{2\sqrt{(ax+b)^3}}{3a}$$ <<*>>= )clear all @@ -687,7 +687,7 @@ result=nn/sqrt(mm) and this reduces to $\sqrt{ax+b}$ \section{\cite{1}:14.90~~~~~$\displaystyle\int{x\sqrt{ax+b}~dx}$} -$$\int{x\sqrt{ax+b}~dx}= +$$\int{x\sqrt{ax+b}}= \frac{2(3ax-2b)}{15a^2}~\sqrt{(ax+b)^3}$$ <<*>>= )clear all @@ -792,7 +792,7 @@ t2*sqrt(t2)-sqrt(t2^3) @ \section{\cite{1}:14.91~~~~~$\displaystyle\int{x^2\sqrt{ax+b}~dx}$} -$$\int{x^2\sqrt{ax+b}~dx}= +$$\int{x^2\sqrt{ax+b}}= \frac{2(15a^2x^2-12abx+8b^2)}{105a^2}~\sqrt{(a+bx)^3}$$ Note: the sqrt term is almost certainly $\sqrt{(ax+b)}$ <<*>>= @@ -844,8 +844,8 @@ cc:=aa-bb @ \section{\cite{1}:14.92~~~~~$\displaystyle\int{\frac{\sqrt{ax+b}}{x}~dx}$} -$$\int{\frac{\sqrt{ax+b}}{x}~dx}= -2\sqrt{ax+b}+b~\int{\frac{dx}{x\sqrt{ax+b}}}$$ +$$\int{\frac{\sqrt{ax+b}}{x}}= +2\sqrt{ax+b}+b~\int{\frac{1}{x\sqrt{ax+b}}}$$ <<*>>= )clear all @@ -977,8 +977,8 @@ cc22:=bb2-aa.2 @ \section{\cite{1}:14.93~~~~~$\displaystyle\int{\frac{\sqrt{ax+b}}{x^2}~dx}$} -$$\int{\frac{\sqrt{ax+b}}{x^2}~dx}= --\frac{\sqrt{ax+b}}{x}+\frac{a}{2}~\int{\frac{dx}{x\sqrt{ax+b}}}$$ +$$\int{\frac{\sqrt{ax+b}}{x^2}}= +-\frac{\sqrt{ax+b}}{x}+\frac{a}{2}~\int{\frac{1}{x\sqrt{ax+b}}}$$ <<*>>= )clear all @@ -1101,9 +1101,9 @@ cc22:=bb2-aa.2 @ \section{\cite{1}:14.94~~~~~$\displaystyle\int{\frac{x^m}{\sqrt{ax+b}}~dx}$} -$$\int{\frac{x^m}{\sqrt{ax+b}}~dx}= +$$\int{\frac{x^m}{\sqrt{ax+b}}}= \frac{2x^m\sqrt{ax+b}}{(2m+1)a}-\frac{2mb}{(2m+1)a} -~\int{\frac{x^{m-1}}{\sqrt{ax+b}}~dx}$$ +~\int{\frac{x^{m-1}}{\sqrt{ax+b}}}$$ <<*>>= )clear all @@ -1121,9 +1121,9 @@ aa:=integrate(x^m/sqrt(a*x+b),x) @ \section{\cite{1}:14.95~~~~~$\displaystyle\int{\frac{dx}{x^m\sqrt{ax+b}}}$} -$$\int{\frac{dx}{x^m\sqrt{ax+b}}}= +$$\int{\frac{1}{x^m\sqrt{ax+b}}}= -\frac{\sqrt{ax+b}}{(m-1)bx^{m-1}}-\frac{(2m-3)a}{(2m-2)b} -~\int{\frac{dx}{x^{m-1}\sqrt{ax+b}}}$$ +~\int{\frac{1}{x^{m-1}\sqrt{ax+b}}}$$ <<*>>= )clear all @@ -1141,9 +1141,9 @@ aa:=integrate(1/(x^m*sqrt(a*x+b)),x) @ \section{\cite{1}:14.96~~~~~$\displaystyle\int{x^m\sqrt{ax+b}~dx}$} -$$\int{x^m\sqrt{ax+b}~dx}= +$$\int{x^m\sqrt{ax+b}}= \frac{2x^m}{(2m+3)a}(ax+b)^{3/2} --\frac{2mb}{(2m+3)a}~\int{x^{m-1}\sqrt{ax+b}~dx}$$ +-\frac{2mb}{(2m+3)a}~\int{x^{m-1}\sqrt{ax+b}}$$ <<*>>= )clear all @@ -1160,9 +1160,9 @@ aa:=integrate(x^m*sqrt(a*x+b),x) @ \section{\cite{1}:14.97~~~~~$\displaystyle\int{\frac{\sqrt{ax+b}}{x^m}~dx}$} -$$\int{\frac{\sqrt{ax+b}}{x^m}~dx}= +$$\int{\frac{\sqrt{ax+b}}{x^m}}= -\frac{\sqrt{ax+b}}{(m-1)x^{m-1}} -+\frac{a}{2(m-1)}~\int{\frac{dx}{x^{m-1}\sqrt{ax+b}}}$$ ++\frac{a}{2(m-1)}~\int{\frac{1}{x^{m-1}\sqrt{ax+b}}}$$ <<*>>= )clear all @@ -1180,9 +1180,9 @@ aa:=integrate(sqrt(a*x+b)/x^m,x) @ \section{\cite{1}:14.98~~~~~$\displaystyle\int{\frac{\sqrt{ax+b}}{x^m}~dx}$} -$$\int{\frac{\sqrt{ax+b}}{x^m}~dx}= +$$\int{\frac{\sqrt{ax+b}}{x^m}}= \frac{-(ax+b)^{3/2}}{(m-1)bx^{m-1}} --\frac{(2m-5)a}{(2m-2)b}~\int{\frac{\sqrt{ax+b}}{x^{m-1}}~dx}$$ +-\frac{(2m-5)a}{(2m-2)b}~\int{\frac{\sqrt{ax+b}}{x^{m-1}}}$$ Note: 14.98 is the same as 14.97 <<*>>= )clear all @@ -1201,7 +1201,7 @@ aa:=integrate(sqrt(a*x+b)/x^m,x) @ \section{\cite{1}:14.99~~~~~$\displaystyle\int{(ax+b)^{m/2}~dx}$} -$$\int{(ax+b)^{m/2}~dx}= +$$\int{(ax+b)^{m/2}}= \frac{2(ax+b)^{(m+2)/2}}{a(m+2)}$$ <<*>>= )clear all @@ -1249,7 +1249,7 @@ cc:=aa-bb @ \section{\cite{1}:14.100~~~~~$\displaystyle\int{x(ax+b)^{m/2}~dx}$} -$$\int{x(ax+b)^{m/2}~dx}= +$$\int{x(ax+b)^{m/2}}= \frac{2(ax+b)^{(m+4)/2}}{a^2(m+4)} -\frac{2b(ax+b)^{(m+2)/2}}{a^2(m+2)}$$ <<*>>= @@ -1307,7 +1307,7 @@ cc:=aa-bb @ \section{\cite{1}:14.101~~~~~$\displaystyle\int{x^2(ax+b)^{m/2}~dx}$} -$$\int{x^2(ax+b)^{m/2}~dx}= +$$\int{x^2(ax+b)^{m/2}}= \frac{2(ax+b)^{(m+6)/2}}{a^3(m+6)} -\frac{4b(ax+b)^{(m+4)/2}}{a^3(m+4)} +\frac{2b^2(ax+b)^{(m+2)/2}}{a^3(m+2)}$$ @@ -1386,9 +1386,9 @@ cc:=aa-bb @ \section{\cite{1}:14.102~~~~~$\displaystyle\int{\frac{(ax+b)^{m/2}}{x}~dx}$} -$$\int{\frac{(ax+b)^{m/2}}{x}~dx}= +$$\int{\frac{(ax+b)^{m/2}}{x}}= \frac{2(ax+b)^{m/2}}{m} -+b~\int{\frac{(ax+b)^{(m-2)/2}}{x}~dx}$$ ++b~\int{\frac{(ax+b)^{(m-2)/2}}{x}}$$ <<*>>= )clear all @@ -1407,9 +1407,9 @@ aa:=integrate((a*x+b)^(m/2)/x,x) @ \section{\cite{1}:14.103~~~~~$\displaystyle \int{\frac{(ax+b)^{m/2}}{x^2}~dx}$} -$$\int{\frac{(ax+b)^{m/2}}{x^2}~dx}= +$$\int{\frac{(ax+b)^{m/2}}{x^2}}= -\frac{(ax+b)^{(m+2)/2}}{bx} -+\frac{ma}{2b}~\int{\frac{(ax+b)^{m/2}}{x}~dx}$$ ++\frac{ma}{2b}~\int{\frac{(ax+b)^{m/2}}{x}}$$ <<*>>= )clear all @@ -1429,9 +1429,9 @@ aa:=integrate((a*x+b)^(m/2)/x^2,x) @ \section{\cite{1}:14.104~~~~~$\displaystyle \int{\frac{dx}{x(ax+b)^{m/2}}}$} -$$\int{\frac{dx}{x(ax+b)^{m/2}}}= +$$\int{\frac{1}{x(ax+b)^{m/2}}}= \frac{2}{(m-2)b(ax+b)^{(m-2)/2}} -+\frac{1}{b}~\int{\frac{dx}{x(ax+b)^{(m-2)/2}}}$$ ++\frac{1}{b}~\int{\frac{1}{x(ax+b)^{(m-2)/2}}}$$ <<*>>= )clear all diff --git a/src/input/schaum3.input.pamphlet b/src/input/schaum3.input.pamphlet index e273509..e1e030d 100644 --- a/src/input/schaum3.input.pamphlet +++ b/src/input/schaum3.input.pamphlet @@ -8,7 +8,7 @@ \tableofcontents \eject \section{\cite{1}:14.105~~~~~$\displaystyle\int{\frac{dx}{(ax+b)(px+q)}}$} -$$\int{\frac{dx}{(ax+b)(px+q)}}= +$$\int{\frac{1}{(ax+b)(px+q)}}= \frac{1}{bp-aq}~\ln\left(\frac{px+q}{ax+b}\right)$$ <<*>>= )spool schaum3.output @@ -54,7 +54,7 @@ cc:=aa-bb @ \section{\cite{1}:14.106~~~~~$\displaystyle\int{\frac{x~dx}{(ax+b)(px+q)}}$} -$$\int{\frac{x~dx}{(ax+b)(px+q)}}= +$$\int{\frac{x}{(ax+b)(px+q)}}= \frac{1}{bp-aq}\left\{\frac{b}{a}~\ln(ax+b)-\frac{q}{p}~\ln(px+q)\right\}$$ <<*>>= )clear all @@ -93,7 +93,7 @@ cc:=aa-bb @ \section{\cite{1}:14.107~~~~~$\displaystyle\int{\frac{dx}{(ax+b)^2(px+q)}}$} -$$\int{\frac{dx}{(ax+b)^2(px+q)}}= +$$\int{\frac{1}{(ax+b)^2(px+q)}}= \frac{1}{bp-aq} \left\{\frac{1}{ax+b}+ \frac{p}{bp-aq}~\ln\left(\frac{px+q}{ax+b}\right)\right\}$$ @@ -141,7 +141,7 @@ cc:=aa-bb @ \section{\cite{1}:14.108~~~~~$\displaystyle\int{\frac{x~dx}{(ax+b)^2(px+q)}}$} -$$\int{\frac{x~dx}{(ax+b)^2(px+q)}}= +$$\int{\frac{x}{(ax+b)^2(px+q)}}= \frac{1}{bp-aq} \left\{\frac{q}{bp-aq} ~\ln\left(\frac{ax+b}{px+q}\right)-\frac{b}{a(ax+b)}\right\}$$ @@ -192,7 +192,7 @@ cc:=aa-bb \section{\cite{1}:14.109~~~~~$\displaystyle \int{\frac{x^2~dx}{(ax+b)^2(px+q)}}$} -$$\int{\frac{x^2~dx}{(ax+b)^2(px+q)}}=$$ +$$\int{\frac{x^2}{(ax+b)^2(px+q)}}=$$ $$\frac{b^2}{(bp-aq)a^2(ax+b)}+\frac{1}{(bp-aq)^2} \left\{\frac{q^2}{p}~\ln(px+q)+\frac{b(bp-2aq)}{a^2}~\ln(ax+b)\right\}$$ <<*>>= @@ -243,10 +243,10 @@ cc:=aa-bb @ \section{\cite{1}:14.110~~~~~$\displaystyle\int{\frac{dx}{(ax+b)^m(px+q)^n}}$} -$$\int{\frac{dx}{(ax+b)^m(px+q)^n}}=$$ +$$\int{\frac{1}{(ax+b)^m(px+q)^n}}=$$ $$\frac{-1}{(n-1)(bp-aq)} \left\{\frac{1}{(ax+b)^{m-1}(px+q)^{n-1}}+ -a(m+n-2)~\int{\frac{dx}{(ax+b)^m(px+q)^{n-1}}}\right\}$$ +a(m+n-2)~\int{\frac{1}{(ax+b)^m(px+q)^{n-1}}}\right\}$$ <<*>>= )clear all @@ -331,7 +331,7 @@ cc:=aa-bb @ \section{\cite{1}:14.111~~~~~$\displaystyle\int{\frac{ax+b}{px+q}~dx}$} -$$\int{\frac{ax+b}{px+q}~dx}=\frac{ax}{p}+\frac{bp-aq}{p^2}~\ln(px+q)$$ +$$\int{\frac{ax+b}{px+q}}=\frac{ax}{p}+\frac{bp-aq}{p^2}~\ln(px+q)$$ <<*>>= )clear all @@ -369,15 +369,15 @@ cc:=aa-bb @ \section{\cite{1}:14.112~~~~~$\displaystyle\int{\frac{(ax+b)^m}{(px+q)^n}~dx}$} -$$\int{\frac{(ax+b)^m}{(px+q)^n}~dx}=\left\{ +$$\int{\frac{(ax+b)^m}{(px+q)^n}}=\left\{ \begin{array}{c} \frac{-1}{(n-1)(bp-aq)} \left\{\frac{(ax+b)^{m+1}}{(px+q)^{n-1}}+(n-m-2)a -\int{\frac{(ax+b)^m}{(px+q)^{n-1}}}~dx\right\}\\ +\int{\frac{(ax+b)^m}{(px+q)^{n-1}}}\right\}\\ \frac{-1}{(n-m-1)p}+\left\{\frac{(ax+b)^m}{(px+q)^{n-1}}+m(bp-aq) -\int{\frac{(ax+b)^{m-1}}{(px+q)^n}}~dx\right\}\\ +\int{\frac{(ax+b)^{m-1}}{(px+q)^n}}\right\}\\ \frac{-1}{(n-1)p}\left\{\frac{(ax+b)^m}{(px+q)^{n-1}}-ma -\int{\frac{(ax+b)^{m-1}}{(px+q)^{n-1}}}~dx\right\} +\int{\frac{(ax+b)^{m-1}}{(px+q)^{n-1}}}\right\} \end{array} \right.$$ <<*>>= diff --git a/src/input/schaum4.input.pamphlet b/src/input/schaum4.input.pamphlet index b57e857..efd9edc 100644 --- a/src/input/schaum4.input.pamphlet +++ b/src/input/schaum4.input.pamphlet @@ -30,8 +30,8 @@ aa:=integrate((p*x+q)/sqrt(a*x+b),x) @ \section{\cite{1}:14.114~~~~~$\displaystyle -\int{\frac{dx}{(px+q)\sqrt{ax+b}}}~dx$} -$$\int{\frac{dx}{(px+q)\sqrt{ax+b}}}= +\int{\frac{dx}{(px+q)\sqrt{ax+b}}}$} +$$\int{\frac{1}{(px+q)\sqrt{ax+b}}}= \left\{ \begin{array}{l} \frac{1}{\sqrt{bp-aq}\sqrt{p}}\ln\left( @@ -117,7 +117,7 @@ aa:=integrate(sqrt(a*x+b)/(p*x+q),x) \section{\cite{1}:14.116~~~~~$\displaystyle\int{(px+b)^n\sqrt{ax+b}}~dx$} $$\int{(px+b)^n\sqrt{ax+b}}= \frac{2(px+q)^{n+1}\sqrt{ax+b}}{(2n+3)p}+\frac{bp-aq}{(2n+3)p} -\int{\frac{(px+q)^n}{\sqrt{ax+b}}}~dx$$ +\int{\frac{(px+q)^n}{\sqrt{ax+b}}}$$ <<*>>= )clear all @@ -136,10 +136,10 @@ aa:=integrate((p*x+q)^n*sqrt(a*x+b),x) \section{\cite{1}:14.117~~~~~$\displaystyle \int{\frac{dx}{(px+b)^n\sqrt{ax+b}}}$} -$$\int{\frac{dx}{(px+b)^n\sqrt{ax+b}}}= +$$\int{\frac{1}{(px+b)^n\sqrt{ax+b}}}= \frac{\sqrt{ax+b}}{(n-1)(aq-bp)(px+q)^{n-1}}+ \frac{(2n-3)a}{2(n-1)(aq-bp)} -\int{\frac{dx}{(px+q)^{n-1}\sqrt{ax+b}}}$$ +\int{\frac{1}{(px+q)^{n-1}\sqrt{ax+b}}}$$ <<*>>= )clear all @@ -183,7 +183,7 @@ aa:=integrate((p*x+q)^n/sqrt(a*x+b),x) \int{\frac{\sqrt{ax+b}}{(px+q)^n}}~dx$} $$\int{\frac{\sqrt{ax+b}}{(px+q)^n}}= \frac{-\sqrt{ax+b}}{(n-1)p(px+q)^{n-1}}+ -\frac{a}{2(n-1)p}\int{\frac{dx}{(px+q)^{n-1}\sqrt{ax+b}}}$$ +\frac{a}{2(n-1)p}\int{\frac{1}{(px+q)^{n-1}\sqrt{ax+b}}}$$ <<*>>= )clear all diff --git a/src/input/schaum5.input.pamphlet b/src/input/schaum5.input.pamphlet index a784b92..f8bccc8 100644 --- a/src/input/schaum5.input.pamphlet +++ b/src/input/schaum5.input.pamphlet @@ -9,7 +9,7 @@ \eject \section{\cite{1}:14.120~~~~~$\displaystyle \int{\frac{dx}{\sqrt{(ax+b)(px+q)}}}$} -$$\int{\frac{dx}{\sqrt{(ax+b)(px+q)}}}= +$$\int{\frac{1}{\sqrt{(ax+b)(px+q)}}}= \left\{ \begin{array}{l} \frac{2}{\sqrt{ap}}\ln\left(\sqrt{a(px+q)}+\sqrt{p(ax+b)}\right)\\ @@ -57,9 +57,9 @@ aa:=integrate(1/sqrt((a*x+b)*(p*x+q)),x) \section{\cite{1}:14.121~~~~~$\displaystyle \int{\frac{x~dx}{\sqrt{(ax+b)(px+q)}}}$} -$$\int{\frac{x~dx}{\sqrt{(ax+b)(px+q)}}}= +$$\int{\frac{x}{\sqrt{(ax+b)(px+q)}}}= \frac{\sqrt{(ax+b)(px+q)}}{ap}-\frac{bp+aq}{2ap} -\int{\frac{dx}{\sqrt{(ax+b)(px+q)}}} +\int{\frac{1}{\sqrt{(ax+b)(px+q)}}} $$ <<*>>= )clear all @@ -138,7 +138,7 @@ aa:=integrate(x/sqrt((a*x+b)*(p*x+q)),x) \section{\cite{1}:14.122~~~~~$\displaystyle\int{\sqrt{(ax+b)(px+q)}}~dx$} $$\int{\sqrt{(ax+b)(px+q)}}= \frac{2apx+bp+aq}{4ap}\sqrt{(ax+b)(px+q)}- -\frac{(bp-aq)^2}{8ap}\int{\frac{dx}{\sqrt{(ax+b)(px+q)}}} +\frac{(bp-aq)^2}{8ap}\int{\frac{1}{\sqrt{(ax+b)(px+q)}}} $$ <<*>>= )clear all @@ -295,7 +295,7 @@ aa:=integrate(sqrt((a*x+b)*(p*x+q)),x) \section{\cite{1}:14.123~~~~~$\displaystyle\int{\sqrt{\frac{px+q}{ax+b}}}~dx$} $$\int{\sqrt{\frac{px+q}{ax+b}}}= \frac{\sqrt{(ax+b)(px+q)}}{a}+\frac{aq-bp}{2a} -\int{\frac{dx}{\sqrt{(ax+b)(px+q)}}} +\int{\frac{1}{\sqrt{(ax+b)(px+q)}}} $$ <<*>>= )clear all @@ -336,7 +336,7 @@ aa:=integrate(sqrt((p*x+q)/(a*x+b)),x) \section{\cite{1}:14.124~~~~~$\displaystyle \int{\frac{dx}{(px+q)\sqrt{(ax+b)(px+q)}}}~dx$} -$$\int{\frac{dx}{(px+q)\sqrt{(ax+b)(px+q)}}}= +$$\int{\frac{1}{(px+q)\sqrt{(ax+b)(px+q)}}}= \frac{2\sqrt{ax+b}}{(aq-bp)\sqrt{px+q}} $$ <<*>>= diff --git a/src/input/schaum6.input.pamphlet b/src/input/schaum6.input.pamphlet index 1a4b430..9a08dc6 100644 --- a/src/input/schaum6.input.pamphlet +++ b/src/input/schaum6.input.pamphlet @@ -8,7 +8,7 @@ \tableofcontents \eject \section{\cite{1}:14.125~~~~~$\displaystyle\int{\frac{dx}{x^2+a^2}}$} -$$\int{\frac{dx}{x^2+a^2}}=\frac{1}{a}\tan^{-1}\frac{x}{a}$$ +$$\int{\frac{1}{x^2+a^2}}=\frac{1}{a}\tan^{-1}\frac{x}{a}$$ <<*>>= )spool schaum6.output )set message test on @@ -29,7 +29,7 @@ aa:=integrate(1/(x^2+a^2),x) @ \section{\cite{1}:14.126~~~~~$\displaystyle\int{\frac{x~dx}{x^2+a^2}}$} -$$\int{\frac{x~dx}{x^2+a^2}}=\frac{1}{2}\ln(x^2+a^2)$$ +$$\int{\frac{x}{x^2+a^2}}=\frac{1}{2}\ln(x^2+a^2)$$ <<*>>= )clear all @@ -46,7 +46,7 @@ aa:=integrate(x/(x^2+a^2),x) @ \section{\cite{1}:14.127~~~~~$\displaystyle\int{\frac{x^2~dx}{x^2+a^2}}$} -$$\int{\frac{x^2~dx}{x^2+a^2}}=x-a\tan^{-1}\frac{x}{a}$$ +$$\int{\frac{x^2}{x^2+a^2}}=x-a\tan^{-1}\frac{x}{a}$$ <<*>>= )clear all @@ -62,7 +62,7 @@ aa:=integrate(x^2/(x^2+a^2),x) @ \section{\cite{1}:14.128~~~~~$\displaystyle\int{\frac{x^3~dx}{x^2+a^2}}$} -$$\int{\frac{x^3~dx}{x^2+a^2}}=\frac{x^2}{2}-\frac{a^2}{2}\ln(x^2+a^2)$$ +$$\int{\frac{x^3}{x^2+a^2}}=\frac{x^2}{2}-\frac{a^2}{2}\ln(x^2+a^2)$$ <<*>>= )clear all @@ -79,8 +79,8 @@ aa:=integrate(x^3/(x^2+a^2),x) --E @ -\section{\cite{1}:14.129~~~~~$\displaystyle\int{\frac{dx}{x(x^2+a^2)}}~dx$} -$$\int{\frac{dx}{x(x^2+a^2)}}= +\section{\cite{1}:14.129~~~~~$\displaystyle\int{\frac{dx}{x(x^2+a^2)}}$} +$$\int{\frac{1}{x(x^2+a^2)}}= \frac{1}{2a^2}\ln\left(\frac{x^2}{x^2+a^2}\right) $$ <<*>>= @@ -99,8 +99,8 @@ aa:=integrate(1/(x*(x^2+a^2)),x) --E @ -\section{\cite{1}:14.130~~~~~$\displaystyle\int{\frac{dx}{x^2(x^2+a^2)}}~dx$} -$$\int{\frac{dx}{x^2(x^2+a^2)}}= +\section{\cite{1}:14.130~~~~~$\displaystyle\int{\frac{dx}{x^2(x^2+a^2)}}$} +$$\int{\frac{1}{x^2(x^2+a^2)}}= -\frac{1}{a^2x}-\frac{1}{a^3}\tan^{-1}\frac{x}{a} $$ <<*>>= @@ -120,8 +120,8 @@ aa:=integrate(1/(x^2*(x^2+a^2)),x) --E @ -\section{\cite{1}:14.131~~~~~$\displaystyle\int{\frac{dx}{x^3(x^2+a^2)}}~dx$} -$$\int{\frac{dx}{x^3(x^2+a^2)}}= +\section{\cite{1}:14.131~~~~~$\displaystyle\int{\frac{dx}{x^3(x^2+a^2)}}$} +$$\int{\frac{1}{x^3(x^2+a^2)}}= -\frac{1}{2a^2x^2}-\frac{1}{2a^4}\ln\left(\frac{x^2}{x^2+a^2}\right) $$ <<*>>= @@ -140,8 +140,8 @@ aa:=integrate(1/(x^3*(x^2+a^2)),x) --E @ -\section{\cite{1}:14.132~~~~~$\displaystyle\int{\frac{dx}{(x^2+a^2)^2}}~dx$} -$$\int{\frac{dx}{(x^2+a^2)^2}}= +\section{\cite{1}:14.132~~~~~$\displaystyle\int{\frac{dx}{(x^2+a^2)^2}}$} +$$\int{\frac{1}{(x^2+a^2)^2}}= \frac{x}{2a^2(x^2+a^2)}+\frac{1}{2a^3}\tan^{-1}\frac{x}{a} $$ <<*>>= @@ -161,8 +161,8 @@ aa:=integrate(1/((x^2+a^2)^2),x) --E @ -\section{\cite{1}:14.133~~~~~$\displaystyle\int{\frac{x~dx}{(x^2+a^2)^2}}~dx$} -$$\int{\frac{x~dx}{(x^2+a^2)^2}}= +\section{\cite{1}:14.133~~~~~$\displaystyle\int{\frac{x~dx}{(x^2+a^2)^2}}$} +$$\int{\frac{x}{(x^2+a^2)^2}}= \frac{-1}{2(x^2+a^2)} $$ <<*>>= @@ -180,8 +180,8 @@ aa:=integrate(x/((x^2+a^2)^2),x) --E @ -\section{\cite{1}:14.134~~~~~$\displaystyle\int{\frac{x^2dx}{(x^2+a^2)^2}}~dx$} -$$\int{\frac{x^2dx}{(x^2+a^2)^2}}= +\section{\cite{1}:14.134~~~~~$\displaystyle\int{\frac{x^2dx}{(x^2+a^2)^2}}$} +$$\int{\frac{x^2}{(x^2+a^2)^2}}= \frac{-x}{2(x^2+a^2)}+\frac{1}{2a}\tan^{-1}\frac{x}{a} $$ <<*>>= @@ -201,8 +201,8 @@ aa:=integrate(x^2/((x^2+a^2)^2),x) --E @ -\section{\cite{1}:14.135~~~~~$\displaystyle\int{\frac{x^3dx}{(x^2+a^2)^2}}~dx$} -$$\int{\frac{x^3dx}{(x^2+a^2)^2}}= +\section{\cite{1}:14.135~~~~~$\displaystyle\int{\frac{x^3dx}{(x^2+a^2)^2}}$} +$$\int{\frac{x^3}{(x^2+a^2)^2}}= \frac{a^2}{2(x^2+a^2)}+\frac{1}{2}\ln(x^2+a^2) $$ <<*>>= @@ -221,8 +221,8 @@ aa:=integrate(x^3/((x^2+a^2)^2),x) --E @ -\section{\cite{1}:14.136~~~~~$\displaystyle\int{\frac{dx}{x(x^2+a^2)^2}}~dx$} -$$\int{\frac{dx}{x(x^2+a^2)^2}}= +\section{\cite{1}:14.136~~~~~$\displaystyle\int{\frac{dx}{x(x^2+a^2)^2}}$} +$$\int{\frac{1}{x(x^2+a^2)^2}}= \frac{1}{2a^2(x^2+a^2)}+\frac{1}{2a^4}\ln\left(\frac{x^2}{x^2+a^2}\right) $$ <<*>>= @@ -241,8 +241,8 @@ aa:=integrate(1/(x*(x^2+a^2)^2),x) --E @ -\section{\cite{1}:14.137~~~~~$\displaystyle\int{\frac{dx}{x^2(x^2+a^2)^2}}~dx$} -$$\int{\frac{dx}{x^2(x^2+a^2)^2}}= +\section{\cite{1}:14.137~~~~~$\displaystyle\int{\frac{dx}{x^2(x^2+a^2)^2}}$} +$$\int{\frac{1}{x^2(x^2+a^2)^2}}= -\frac{1}{a^4x}-\frac{x}{2a^4(x^2+a^2)}-\frac{3}{2a^5}\tan^{-1}\frac{x}{a} $$ <<*>>= @@ -262,8 +262,8 @@ aa:=integrate(1/((x^2+a^2)^2),x) --E @ -\section{\cite{1}:14.138~~~~~$\displaystyle\int{\frac{dx}{x^3(x^2+a^2)^2}}~dx$} -$$\int{\frac{dx}{x^3(x^2+a^2)^2}}= +\section{\cite{1}:14.138~~~~~$\displaystyle\int{\frac{dx}{x^3(x^2+a^2)^2}}$} +$$\int{\frac{1}{x^3(x^2+a^2)^2}}= -\frac{1}{2a^4x^2}-\frac{1}{2a^4(x^2+a^2)}- \frac{1}{a^6}\ln\left(\frac{x^2}{x^2+a^2}\right) $$ @@ -283,10 +283,10 @@ aa:=integrate(1/(x^3*(x^2+a^2)^2),x) --E @ -\section{\cite{1}:14.139~~~~~$\displaystyle\int{\frac{dx}{(x^2+a^2)^n}}~dx$} -$$\int{\frac{dx}{(x^2+a^2)^n}}= +\section{\cite{1}:14.139~~~~~$\displaystyle\int{\frac{dx}{(x^2+a^2)^n}}$} +$$\int{\frac{1}{(x^2+a^2)^n}}= \frac{x}{2(n-1)a^2(x^2+a^2)^{n-1}}+\frac{2n-3}{(2n-2)a^2} -\int{\frac{dx}{(x^2+a^2)^{n-1}}} +\int{\frac{1}{(x^2+a^2)^{n-1}}} $$ <<*>>= )clear all @@ -304,8 +304,8 @@ aa:=integrate(1/((x^2+a^2)^n),x) --E @ -\section{\cite{1}:14.140~~~~~$\displaystyle\int{\frac{x~dx}{(x^2+a^2)^n}}~dx$} -$$\int{\frac{x~dx}{(x^2+a^2)^n}}= +\section{\cite{1}:14.140~~~~~$\displaystyle\int{\frac{x~dx}{(x^2+a^2)^n}}$} +$$\int{\frac{x}{(x^2+a^2)^n}}= \frac{-1}{2(n-1)(x^2+a^2)^{n-1}} $$ <<*>>= @@ -325,10 +325,10 @@ aa:=integrate(x/((x^2+a^2)^n),x) --E @ -\section{\cite{1}:14.141~~~~~$\displaystyle\int{\frac{dx}{x(x^2+a^2)^n}}~dx$} -$$\int{\frac{dx}{x(x^2+a^2)^n}}= +\section{\cite{1}:14.141~~~~~$\displaystyle\int{\frac{dx}{x(x^2+a^2)^n}}$} +$$\int{\frac{1}{x(x^2+a^2)^n}}= \frac{1}{2(n-1)a^2(x^2+a^2)^{n-1}}+\frac{1}{a^2} -\int{\frac{dx}{x(x^2+a^2)^{n-1}}} +\int{\frac{1}{x(x^2+a^2)^{n-1}}} $$ <<*>>= )clear all @@ -346,10 +346,10 @@ aa:=integrate(1/(x*(x^2+a^2)^n),x) --E @ -\section{\cite{1}:14.142~~~~~$\displaystyle\int{\frac{x^mdx}{(x^2+a^2)^n}}~dx$} -$$\int{\frac{x^mdx}{(x^2+a^2)^n}}= -\int{\frac{x^{m-2}dx}{(x^2+a^2)^{n-1}}} - -a^2\int{\frac{x^{m-2}dx}{(x^2+a^2)^n}} +\section{\cite{1}:14.142~~~~~$\displaystyle\int{\frac{x^mdx}{(x^2+a^2)^n}}$} +$$\int{\frac{x^m}{(x^2+a^2)^n}}= +\int{\frac{x^{m-2}}{(x^2+a^2)^{n-1}}} - +a^2\int{\frac{x^{m-2}}{(x^2+a^2)^n}} $$ <<*>>= )clear all @@ -367,10 +367,10 @@ aa:=integrate(x^m/((x^2+a^2)^n),x) --E @ -\section{\cite{1}:14.143~~~~~$\displaystyle\int{\frac{dx}{x^m(x^2+a^2)^n}}~dx$} -$$\int{\frac{dx}{x^m(x^2+a^2)^n}}= -\frac{1}{a^2}\int{\frac{dx}{x^m(x^2+a^2)^{n-1}}}- -\frac{1}{a^2}\int{\frac{dx}{x^{m-2}(x^2+a^2)^n}} +\section{\cite{1}:14.143~~~~~$\displaystyle\int{\frac{dx}{x^m(x^2+a^2)^n}}$} +$$\int{\frac{1}{x^m(x^2+a^2)^n}}= +\frac{1}{a^2}\int{\frac{1}{x^m(x^2+a^2)^{n-1}}}- +\frac{1}{a^2}\int{\frac{1}{x^{m-2}(x^2+a^2)^n}} $$ <<*>>= )clear all diff --git a/src/input/schaum7.input.pamphlet b/src/input/schaum7.input.pamphlet new file mode 100644 index 0000000..9226da0 --- /dev/null +++ b/src/input/schaum7.input.pamphlet @@ -0,0 +1,395 @@ +\documentclass{article} +\usepackage{axiom} +\begin{document} +\title{\$SPAD/input schaum7.input} +\author{Timothy Daly} +\maketitle +\eject +\tableofcontents +\eject +\section{\cite{1}:14.144~~~~~$\displaystyle\int{\frac{dx}{x^2-a^2}}$} +$$\int{\frac{1}{x^2-a^2}}=\frac{1}{2a}\ln\left(\frac{x-a}{x+a}\right)$$ +$$\int{\frac{1}{x^2-a^2}}=-\frac{1}{a}\coth^{-1}\frac{x}{a}$$ +<<*>>= +)spool schaum7.output +)set message test on +)set message auto off +)clear all + +--S 1 of 19 +aa:=integrate(1/(x^2-a^2),x) +--R +--R +--R - log(x + a) + log(x - a) +--R (1) ------------------------- +--R 2a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.145~~~~~$\displaystyle\int{\frac{x~dx}{x^2-a^2}}$} +$$\int{\frac{x}{x^2-a^2}}=\frac{1}{2}\ln(x^2-a^2)$$ +<<*>>= +)clear all + +--S 2 of 19 +aa:=integrate(x/(x^2-a^2),x) +--R +--R +--R 2 2 +--R log(x - a ) +--R (1) ------------ +--R 2 +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.146~~~~~$\displaystyle\int{\frac{x^2~dx}{x^2-a^2}}$} +$$\int{\frac{x^2}{x^2-a^2}}=x+\frac{a}{2}\ln\left(\frac{x-a}{x+a}\right)$$ +<<*>>= +)clear all + +--S 3 of 19 +aa:=integrate(x^2/(x^2-a^2),x) +--R +--R +--R - a log(x + a) + a log(x - a) + 2x +--R (1) ---------------------------------- +--R 2 +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.147~~~~~$\displaystyle\int{\frac{x^3~dx}{x^2-a^2}}$} +$$\int{\frac{x^3}{x^2-a^2}}=\frac{x^2}{2}+\frac{a^2}{2}\ln(x^2-a^2)$$ + +<<*>>= +)clear all + +--S 4 of 19 +aa:=integrate(x^3/(x^2-a^2),x) +--R +--R +--R 2 2 2 2 +--R a log(x - a ) + x +--R (1) ------------------- +--R 2 +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.148~~~~~$\displaystyle\int{\frac{dx}{x(x^2-a^2)}}$} +$$\int{\frac{1}{x(x^2-a^2)}}= +\frac{1}{2a^2}\ln\left(\frac{x^2-a^2}{x^2}\right) +$$ +<<*>>= +)clear all + +--S 5 of 19 +aa:=integrate(1/(x*(x^2-a^2)),x) +--R +--R +--R 2 2 +--R log(x - a ) - 2log(x) +--R (1) ---------------------- +--R 2 +--R 2a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.149~~~~~$\displaystyle\int{\frac{dx}{x^2(x^2-a^2)}}$} +$$\int{\frac{1}{x^2(x^2-a^2)}}= +\frac{1}{a^2x}+\frac{1}{2a^3}\ln\left(\frac{x-a}{x+a}\right) +$$ +<<*>>= +)clear all + +--S 6 of 19 +aa:=integrate(1/(x^2*(x^2-a^2)),x) +--R +--R +--R - x log(x + a) + x log(x - a) + 2a +--R (1) ---------------------------------- +--R 3 +--R 2a x +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.150~~~~~$\displaystyle\int{\frac{dx}{x^3(x^2-a^2)}}$} +$$\int{\frac{1}{x^3(x^2-a^2)}}= +\frac{1}{2a^2x^2}-\frac{1}{2a^4}\ln\left(\frac{x^2}{x^2-a^2}\right) +$$ +<<*>>= +)clear all + +--S 7 of 19 +aa:=integrate(1/(x^3*(x^2-a^2)),x) +--R +--R +--R 2 2 2 2 2 +--R x log(x - a ) - 2x log(x) + a +--R (1) ------------------------------- +--R 4 2 +--R 2a x +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.151~~~~~$\displaystyle\int{\frac{dx}{(x^2-a^2)^2}}$} +$$\int{\frac{1}{(x^2-a^2)^2}}= +\frac{-x}{2a^2(x^2-a^2)}-\frac{1}{4a^3}\ln\left(\frac{x-a}{x+a}\right) +$$ +<<*>>= +)clear all + +--S 8 of 19 +aa:=integrate(1/((x^2-a^2)^2),x) +--R +--R +--R 2 2 2 2 +--R (x - a )log(x + a) + (- x + a )log(x - a) - 2a x +--R (1) -------------------------------------------------- +--R 3 2 5 +--R 4a x - 4a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.152~~~~~$\displaystyle\int{\frac{x~dx}{(x^2-a^2)^2}}$} +$$\int{\frac{x}{(x^2-a^2)^2}}= +\frac{-1}{2(x^2-a^2)} +$$ +<<*>>= +)clear all + +--S 9 of 19 +aa:=integrate(x/((x^2-a^2)^2),x) +--R +--R +--R 1 +--R (1) - --------- +--R 2 2 +--R 2x - 2a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.153~~~~~$\displaystyle\int{\frac{x^2dx}{(x^2-a^2)^2}}$} +$$\int{\frac{x^2}{(x^2-a^2)^2}}= +\frac{-x}{2(x^2-a^2)}+\frac{1}{4a}\ln\left(\frac{x-a}{x+a}\right) +$$ +<<*>>= +)clear all + +--S 10 of 19 +aa:=integrate(x^2/((x^2-a^2)^2),x) +--R +--R +--R 2 2 2 2 +--R (- x + a )log(x + a) + (x - a )log(x - a) - 2a x +--R (1) -------------------------------------------------- +--R 2 3 +--R 4a x - 4a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.154~~~~~$\displaystyle\int{\frac{x^3dx}{(x^2-a^2)^2}}$} +$$\int{\frac{x^3}{(x^2-a^2)^2}}= +\frac{-a^2}{2(x^2-a^2)}+\frac{1}{2}\ln(x^2-a^2) +$$ +<<*>>= +)clear all + +--S 11 of 19 +aa:=integrate(x^3/((x^2-a^2)^2),x) +--R +--R +--R 2 2 2 2 2 +--R (x - a )log(x - a ) - a +--R (1) -------------------------- +--R 2 2 +--R 2x - 2a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.155~~~~~$\displaystyle\int{\frac{dx}{x(x^2-a^2)^2}}$} +$$\int{\frac{1}{x(x^2-a^2)^2}}= +\frac{-1}{2a^2(x^2-a^2)}+\frac{1}{2a^4}\ln\left(\frac{x^2}{x^2-a^2}\right) +$$ +<<*>>= +)clear all + +--S 12 of 19 +aa:=integrate(1/(x*(x^2-a^2)^2),x) +--R +--R +--R 2 2 2 2 2 2 2 +--R (- x + a )log(x - a ) + (2x - 2a )log(x) - a +--R (1) ------------------------------------------------ +--R 4 2 6 +--R 2a x - 2a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.156~~~~~$\displaystyle\int{\frac{dx}{x^2(x^2-a^2)^2}}$} +$$\int{\frac{1}{x^2(x^2-a^2)^2}}= +-\frac{1}{a^4x}-\frac{x}{2a^4(x^2-a^2)}- +\frac{3}{4a^5}\ln\left(\frac{x-a}{x+a}\right) +$$ +<<*>>= +)clear all + +--S 13 of 19 +aa:=integrate(1/((x^2-a^2)^2),x) +--R +--R +--R 2 2 2 2 +--R (x - a )log(x + a) + (- x + a )log(x - a) - 2a x +--R (1) -------------------------------------------------- +--R 3 2 5 +--R 4a x - 4a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.157~~~~~$\displaystyle\int{\frac{dx}{x^3(x^2-a^2)^2}}$} +$$\int{\frac{1}{x^3(x^2-a^2)^2}}= +-\frac{1}{2a^4x^2}-\frac{1}{2a^4(x^2-a^2)}+ +\frac{1}{a^6}\ln\left(\frac{x^2}{x^2-a^2}\right) +$$ +<<*>>= +)clear all + +--S 14 of 19 +aa:=integrate(1/(x^3*(x^2-a^2)^2),x) +--R +--R +--R 4 2 2 2 2 4 2 2 2 2 4 +--R (- 2x + 2a x )log(x - a ) + (4x - 4a x )log(x) - 2a x + a +--R (1) -------------------------------------------------------------- +--R 6 4 8 2 +--R 2a x - 2a x +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.158~~~~~$\displaystyle\int{\frac{dx}{(x^2-a^2)^n}}$} +$$\int{\frac{1}{(x^2-a^2)^n}}= +\frac{-x}{2(n-1)a^2(x^2-a^2)^{n-1}}- +\frac{2n-3}{(2n-2)a^2}\int{\frac{1}{(x^2-a^2)^{n-1}}} +$$ +<<*>>= +)clear all + +--S 15 of 19 +aa:=integrate(1/((x^2-a^2)^n),x) +--R +--R +--R x +--R ++ 1 +--I (1) | ------------- d%L +--R ++ 2 2 n +--I (- a + %L ) +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.159~~~~~$\displaystyle\int{\frac{x~dx}{(x^2-a^2)^n}}$} +$$\int{\frac{x}{(x^2-a^2)^n}}= +\frac{-1}{2(n-1)(x^2-a^2)^{n-1}} +$$ +<<*>>= +)clear all + +--S 16 of 19 +aa:=integrate(x/((x^2-a^2)^n),x) +--R +--R +--R 2 2 +--R - x + a +--R (1) ------------------------ +--R 2 2 +--R n log(x - a ) +--R (2n - 2)%e +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.160~~~~~$\displaystyle\int{\frac{dx}{x(x^2-a^2)^n}}$} +$$\int{\frac{1}{x(x^2-a^2)^n}}= +\frac{-1}{2(n-1)a^2(x^2-a^2)^{n-1}}- +\frac{1}{a^2}\int{\frac{1}{x(x^2-a^2)^{n-1}}} +$$ +<<*>>= +)clear all + +--S 17 of 19 +aa:=integrate(1/(x*(x^2-a^2)^n),x) +--R +--R +--R x +--R ++ 1 +--I (1) | ---------------- d%L +--R ++ 2 2 n +--I %L (- a + %L ) +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.161~~~~~$\displaystyle\int{\frac{x^mdx}{(x^2-a^2)^n}}$} +$$\int{\frac{x^m}{(x^2-a^2)^n}}= +\int{\frac{x^{m-2}}{(x^2-a^2)^{n-1}}}+ +a^2\int\frac{x^{m-2}}{(x^2-a^2)^n} +$$ +<<*>>= +)clear all + +--S 18 of 19 +aa:=integrate(x^m/((x^2-a^2)^n),x) +--R +--R +--R x m +--I ++ %L +--I (1) | ------------- d%L +--R ++ 2 2 n +--I (- a + %L ) +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.162~~~~~$\displaystyle\int{\frac{dx}{x^m(x^2-a^2)^n}}$} +$$\int{\frac{1}{x^m(x^2-a^2)^n}}= +\frac{1}{a^2}\int{\frac{1}{x^{m-2}(x^2-a^2)^n}}- +\frac{1}{a^2}\int{\frac{1}{x^m(x^2-a^2)^{n-1}}} +$$ +<<*>>= +)clear all + +--S 19 of 19 +aa:=integrate(1/(x^m*(x^2-a^2)^n),x) +--R +--R +--R x +--R ++ 1 +--I (1) | ---------------- d%L +--R ++ 2 2 n m +--I (- a + %L ) %L +--R Type: Union(Expression Integer,...) +--E + +)spool +)lisp (bye) +@ + +\eject +\begin{thebibliography}{99} +\bibitem{1} Spiegel, Murray R. +{\sl Mathematical Handbook of Formulas and Tables}\\ +Schaum's Outline Series McGraw-Hill 1968 p65 +\end{thebibliography} +\end{document} diff --git a/src/input/schaum8.input.pamphlet b/src/input/schaum8.input.pamphlet new file mode 100644 index 0000000..a5c0c7e --- /dev/null +++ b/src/input/schaum8.input.pamphlet @@ -0,0 +1,395 @@ +\documentclass{article} +\usepackage{axiom} +\begin{document} +\title{\$SPAD/input schaum8.input} +\author{Timothy Daly} +\maketitle +\eject +\tableofcontents +\eject +\section{\cite{1}:14.163~~~~~$\displaystyle\int{\frac{dx}{a^2-x^2}}$} +$$\int{\frac{1}{a^2-x^2}}=\frac{1}{2a}\ln\left(\frac{a-x}{a+x}\right)$$ +$$\int{\frac{1}{a^2-x^2}}=-\frac{1}{a}\coth^{-1}\frac{x}{a}$$ +<<*>>= +)spool schaum8.output +)set message test on +)set message auto off +)clear all + +--S 1 of 19 +aa:=integrate(1/(a^2-x^2),x) +--R +--R +--R log(x + a) - log(x - a) +--R (1) ----------------------- +--R 2a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.164~~~~~$\displaystyle\int{\frac{x~dx}{a^2-x^2}}$} +$$\int{\frac{x}{a^2-x^2}}=-\frac{1}{2}\ln(a^2-x^2)$$ +<<*>>= +)clear all + +--S 2 of 19 +aa:=integrate(x/(a^2-x^2),x) +--R +--R +--R 2 2 +--R log(x - a ) +--R (1) - ------------ +--R 2 +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.165~~~~~$\displaystyle\int{\frac{x^2~dx}{a^2-x^2}}$} +$$\int{\frac{x^2}{a^2-x^2}}=-x+\frac{a}{2}\ln\left(\frac{a+x}{a-x}\right)$$ +<<*>>= +)clear all + +--S 3 of 19 +aa:=integrate(x^2/(a^2-x^2),x) +--R +--R +--R a log(x + a) - a log(x - a) - 2x +--R (1) -------------------------------- +--R 2 +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.166~~~~~$\displaystyle\int{\frac{x^3~dx}{a^2-x^2}}$} +$$\int{\frac{x^3}{a^2-x^2}}=-\frac{x^2}{2}-\frac{a^2}{2}\ln(a^2-x^2)$$ + +<<*>>= +)clear all + +--S 4 of 19 +aa:=integrate(x^3/(a^2-x^2),x) +--R +--R +--R 2 2 2 2 +--R - a log(x - a ) - x +--R (1) --------------------- +--R 2 +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.167~~~~~$\displaystyle\int{\frac{dx}{x(a^2-x^2)}}$} +$$\int{\frac{1}{x(a^2-x^2)}}= +\frac{1}{2a^2}\ln\left(\frac{x^2}{a^2-x^2}\right) +$$ +<<*>>= +)clear all + +--S 5 of 19 +aa:=integrate(1/(x*(a^2-x^2)),x) +--R +--R +--R 2 2 +--R - log(x - a ) + 2log(x) +--R (1) ------------------------ +--R 2 +--R 2a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.168~~~~~$\displaystyle\int{\frac{dx}{x^2(a^2-x^2)}}$} +$$\int{\frac{1}{x^2(a^2-x^2)}}= +\frac{1}{a^2x}+\frac{1}{2a^3}\ln\left(\frac{a+x}{a-x}\right) +$$ +<<*>>= +)clear all + +--S 6 of 19 +aa:=integrate(1/(x^2*(a^2-x^2)),x) +--R +--R +--R x log(x + a) - x log(x - a) - 2a +--R (1) -------------------------------- +--R 3 +--R 2a x +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.169~~~~~$\displaystyle\int{\frac{dx}{x^3(a^2-x^2)}}$} +$$\int{\frac{1}{x^3(a^2-x^2)}}= +-\frac{1}{2a^2x^2}+\frac{1}{2a^4}\ln\left(\frac{x^2}{a^2-x^2}\right) +$$ +<<*>>= +)clear all + +--S 7 of 19 +aa:=integrate(1/(x^3*(a^2-x^2)),x) +--R +--R +--R 2 2 2 2 2 +--R - x log(x - a ) + 2x log(x) - a +--R (1) --------------------------------- +--R 4 2 +--R 2a x +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.170~~~~~$\displaystyle\int{\frac{dx}{(a^2-x^2)^2}}$} +$$\int{\frac{1}{(a^2-x^2)^2}}= +\frac{x}{2a^2(a^2-x^2)}+\frac{1}{4a^3}\ln\left(\frac{a+x}{a-x}\right) +$$ +<<*>>= +)clear all + +--S 8 of 19 +aa:=integrate(1/((a^2-x^2)^2),x) +--R +--R +--R 2 2 2 2 +--R (x - a )log(x + a) + (- x + a )log(x - a) - 2a x +--R (1) -------------------------------------------------- +--R 3 2 5 +--R 4a x - 4a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.171~~~~~$\displaystyle\int{\frac{x~dx}{(a^2-x^2)^2}}$} +$$\int{\frac{x}{(a^2-x^2)^2}}= +\frac{1}{2(a^2-x^2)} +$$ +<<*>>= +)clear all + +--S 9 of 19 +aa:=integrate(x/((a^2-x^2)^2),x) +--R +--R +--R 1 +--R (1) - --------- +--R 2 2 +--R 2x - 2a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.172~~~~~$\displaystyle\int{\frac{x^2dx}{(a^2-x^2)^2}}$} +$$\int{\frac{x^2}{(a^2-x^2)^2}}= +\frac{x}{2(a^2-x^2)}-\frac{1}{4a}\ln\left(\frac{a+x}{a-x}\right) +$$ +<<*>>= +)clear all + +--S 10 of 19 +aa:=integrate(x^2/((a^2-x^2)^2),x) +--R +--R +--R 2 2 2 2 +--R (- x + a )log(x + a) + (x - a )log(x - a) - 2a x +--R (1) -------------------------------------------------- +--R 2 3 +--R 4a x - 4a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.173~~~~~$\displaystyle\int{\frac{x^3dx}{(a^2-x^2)^2}}$} +$$\int{\frac{x^3}{(a^2-x^2)^2}}= +\frac{a^2}{2(a^2-x^2)}+\frac{1}{2}\ln(a^2-x^2) +$$ +<<*>>= +)clear all + +--S 11 of 19 +aa:=integrate(x^3/((a^2-x^2)^2),x) +--R +--R +--R 2 2 2 2 2 +--R (x - a )log(x - a ) - a +--R (1) -------------------------- +--R 2 2 +--R 2x - 2a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.174~~~~~$\displaystyle\int{\frac{dx}{x(a^2-x^2)^2}}$} +$$\int{\frac{1}{x(a^2-x^2)^2}}= +\frac{1}{2a^2(a^2-x^2)}+\frac{1}{2a^4}\ln\left(\frac{x^2}{a^2-x^2}\right) +$$ +<<*>>= +)clear all + +--S 12 of 19 +aa:=integrate(1/(x*(a^2-x^2)^2),x) +--R +--R +--R 2 2 2 2 2 2 2 +--R (- x + a )log(x - a ) + (2x - 2a )log(x) - a +--R (1) ------------------------------------------------ +--R 4 2 6 +--R 2a x - 2a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.175~~~~~$\displaystyle\int{\frac{dx}{x^2(a^2-x^2)^2}}$} +$$\int{\frac{1}{x^2(a^2-x^2)^2}}= +-\frac{1}{a^4x}+\frac{x}{2a^4(a^2-x^2)}+ +\frac{3}{4a^5}\ln\left(\frac{a+x}{a-x}\right) +$$ +<<*>>= +)clear all + +--S 13 of 19 +aa:=integrate(1/((a^2-x^2)^2),x) +--R +--R +--R 2 2 2 2 +--R (x - a )log(x + a) + (- x + a )log(x - a) - 2a x +--R (1) -------------------------------------------------- +--R 3 2 5 +--R 4a x - 4a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.176~~~~~$\displaystyle\int{\frac{dx}{x^3(a^2-x^2)^2}}$} +$$\int{\frac{1}{x^3(a^2-x^2)^2}}= +\frac{1}{2a^4x^2}+\frac{1}{2a^4(a^2-x^2)}+ +\frac{1}{a^6}\ln\left(\frac{x^2}{a^2-x^2}\right) +$$ +<<*>>= +)clear all + +--S 14 of 19 +aa:=integrate(1/(x^3*(a^2-x^2)^2),x) +--R +--R +--R 4 2 2 2 2 4 2 2 2 2 4 +--R (- 2x + 2a x )log(x - a ) + (4x - 4a x )log(x) - 2a x + a +--R (1) -------------------------------------------------------------- +--R 6 4 8 2 +--R 2a x - 2a x +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.177~~~~~$\displaystyle\int{\frac{dx}{(a^2-x^2)^n}}$} +$$\int{\frac{1}{(a^2-x^2)^n}}= +\frac{x}{2(n-1)a^2(a^2-x^2)^{n-1}}+ +\frac{2n-3}{(2n-2)a^2}\int{\frac{1}{(a^2-x^2)^{n-1}}} +$$ +<<*>>= +)clear all + +--S 15 of 19 +aa:=integrate(1/((a^2-x^2)^n),x) +--R +--R +--R x +--R ++ 1 +--I (1) | ----------- d%L +--R ++ 2 2 n +--I (a - %L ) +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.178~~~~~$\displaystyle\int{\frac{x~dx}{(a^2-x^2)^n}}$} +$$\int{\frac{x}{(a^2-x^2)^n}}= +\frac{1}{2(n-1)(a^2-x^2)^{n-1}} +$$ +<<*>>= +)clear all + +--S 16 of 19 +aa:=integrate(x/((a^2-x^2)^n),x) +--R +--R +--R 2 2 +--R - x + a +--R (1) -------------------------- +--R 2 2 +--R n log(- x + a ) +--R (2n - 2)%e +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.179~~~~~$\displaystyle\int{\frac{dx}{x(a^2-x^2)^n}}$} +$$\int{\frac{1}{x(a^2-x^2)^n}}= +\frac{1}{2(n-1)a^2(a^2-x^2)^{n-1}}+ +\frac{1}{a^2}\int{\frac{1}{x(a^2-x^2)^{n-1}}} +$$ +<<*>>= +)clear all + +--S 17 of 19 +aa:=integrate(1/(x*(a^2-x^2)^n),x) +--R +--R +--R x +--R ++ 1 +--I (1) | -------------- d%L +--R ++ 2 2 n +--I %L (a - %L ) +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.180~~~~~$\displaystyle\int{\frac{x^mdx}{(a^2-x^2)^n}}$} +$$\int{\frac{x^m}{(a^2-x^2)^n}}= +a^2\int\frac{x^{m-2}}{(a^2-x^2)^n}- +\int{\frac{x^{m-2}}{(a^2-x^2)^{n-1}}} +$$ +<<*>>= +)clear all + +--S 18 of 19 +aa:=integrate(x^m/((a^2-x^2)^n),x) +--R +--R +--R x m +--I ++ %L +--I (1) | ----------- d%L +--R ++ 2 2 n +--I (a - %L ) +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.181~~~~~$\displaystyle\int{\frac{dx}{x^m(a^2-x^2)^n}}$} +$$\int{\frac{1}{x^m(a^2-x^2)^n}}= +\frac{1}{a^2}\int{\frac{1}{x^m(a^2-x^2)^{n-1}}}+ +\frac{1}{a^2}\int{\frac{1}{x^{m-2}(a^2-x^2)^n}} +$$ +<<*>>= +)clear all + +--S 19 of 19 +aa:=integrate(1/(x^m*(a^2-x^2)^n),x) +--R +--R +--R x +--R ++ 1 +--I (1) | -------------- d%L +--R ++ m 2 2 n +--I %L (a - %L ) +--R Type: Union(Expression Integer,...) +--E + +)spool +)lisp (bye) +@ + +\eject +\begin{thebibliography}{99} +\bibitem{1} Spiegel, Murray R. +{\sl Mathematical Handbook of Formulas and Tables}\\ +Schaum's Outline Series McGraw-Hill 1968 p66 +\end{thebibliography} +\end{document} diff --git a/src/input/schaum9.input.pamphlet b/src/input/schaum9.input.pamphlet new file mode 100644 index 0000000..6610811 --- /dev/null +++ b/src/input/schaum9.input.pamphlet @@ -0,0 +1,776 @@ +\documentclass{article} +\usepackage{axiom} +\begin{document} +\title{\$SPAD/input schaum9.input} +\author{Timothy Daly} +\maketitle +\eject +\tableofcontents +\eject +\section{\cite{1}:14.182~~~~~$\displaystyle\int{\frac{dx}{\sqrt{x^2+a^2}}}$} +$$\int{\frac{1}{\sqrt{x^2+a^2}}}=\ln\left(x+\sqrt{x^2+a^2}\right)$$ +$$\int{\frac{1}{\sqrt{x^2+a^2}}}=\sinh^{-1}\frac{x}{a}$$ +<<*>>= +)spool schaum9.output +)set message test on +)set message auto off +)clear all + +--S 1 of 28 +aa:=integrate(1/(sqrt(x^2+a^2)),x) +--R +--R +--R +-------+ +--R | 2 2 +--R (1) - log(\|x + a - x) +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.183~~~~~$\displaystyle\int{\frac{x~dx}{\sqrt{x^2+a^2}}}$} +$$\int{\frac{x}{\sqrt{x^2+a^2}}}=\sqrt{x^2+a^2}$$ +<<*>>= +)clear all + +--S 2 of 28 +aa:=integrate(x/(sqrt(x^2+a^2)),x) +--R +--R +--R +-------+ +--R | 2 2 2 2 +--R - x\|x + a + x + a +--R (1) ----------------------- +--R +-------+ +--R | 2 2 +--R \|x + a - x +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.184~~~~~$\displaystyle +\int{\frac{x^2~dx}{\sqrt{x^2+a^2}}}$} +$$\int{\frac{x^2}{\sqrt{x^2+a^2}}}= +\frac{x\sqrt{x^2+a^2}}{2}-\frac{a^2}{2}\ln\left(x+\sqrt{x^2+a^2}\right) +$$ +<<*>>= +)clear all + +--S 3 of 28 +aa:=integrate(x^2/sqrt(x^2+a^2),x) +--R +--R +--R (1) +--R +-------+ +-------+ +--R 2 | 2 2 2 2 4 | 2 2 +--R (2a x\|x + a - 2a x - a )log(\|x + a - x) +--R + +--R +-------+ +--R 3 2 | 2 2 4 2 2 +--R (- 2x - a x)\|x + a + 2x + 2a x +--R / +--R +-------+ +--R | 2 2 2 2 +--R 4x\|x + a - 4x - 2a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.185~~~~~$\displaystyle +\int{\frac{x^3~dx}{\sqrt{x^2+a^2}}}$} +$$\int{\frac{x^3}{\sqrt{x^2+a^2}}}= +\frac{(x^2+a^2)^{3/2}}{3}-a^2\sqrt{x^2+a^2} +$$ +<<*>>= +)clear all + +--S 4 of 28 +aa:=integrate(x^3/sqrt(x^2+a^2),x) +--R +--R +--R +-------+ +--R 5 2 3 4 | 2 2 6 2 4 4 2 6 +--R (- 4x + 5a x + 6a x)\|x + a + 4x - 3a x - 9a x - 2a +--R (1) ------------------------------------------------------------ +--R +-------+ +--R 2 2 | 2 2 3 2 +--R (12x + 3a )\|x + a - 12x - 9a x +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.186~~~~~$\displaystyle\int{\frac{dx}{x\sqrt{x^2+a^2}}}$} +$$\int{\frac{1}{x\sqrt{x^2+a^2}}}= +-\frac{1}{a}\ln\left(\frac{a+\sqrt{x^2+a^2}}{x}\right) +$$ +<<*>>= +)clear all + +--S 5 of 28 +aa:=integrate(1/(x*sqrt(x^2+a^2)),x) +--R +--R +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R - log(\|x + a - x + a) + log(\|x + a - x - a) +--R (1) --------------------------------------------------- +--R a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.187~~~~~$\displaystyle +\int{\frac{dx}{x^2\sqrt{x^2+a^2}}}$} +$$\int{\frac{1}{x^2\sqrt{x^2+a^2}}}= +-\frac{\sqrt{x^2+a^2}}{a^2x} +$$ +<<*>>= +)clear all + +--S 6 of 28 +aa:=integrate(1/(x^2*sqrt(x^2+a^2)),x) +--R +--R +--R 1 +--R (1) - ---------------- +--R +-------+ +--R | 2 2 2 +--R x\|x + a - x +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.188~~~~~$\displaystyle\int{\frac{dx}{x^3\sqrt{x^2+a^2}}}$} +$$\int{\frac{1}{x^3\sqrt{x^2+a^2}}}= +-\frac{\sqrt{x^2+a^2}}{2a^2x^2}+\frac{1}{2a^3} +\ln\left(\frac{a+\sqrt{x^2+a^2}}{x}\right) +$$ +<<*>>= +)clear all + +--S 7 of 28 +aa:=integrate(1/(x^3*sqrt(x^2+a^2)),x) +--R +--R +--R (1) +--R +-------+ +-------+ +--R 3 | 2 2 4 2 2 | 2 2 +--R (2x \|x + a - 2x - a x )log(\|x + a - x + a) +--R + +--R +-------+ +-------+ +--R 3 | 2 2 4 2 2 | 2 2 +--R (- 2x \|x + a + 2x + a x )log(\|x + a - x - a) +--R + +--R +-------+ +--R 2 3 | 2 2 3 3 +--R (2a x + a )\|x + a - 2a x - 2a x +--R / +--R +-------+ +--R 3 3 | 2 2 3 4 5 2 +--R 4a x \|x + a - 4a x - 2a x +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.189~~~~~$\displaystyle\int{\sqrt{x^2+a^2}}~dx$} +$$\int{\sqrt{x^2+a^2}}= +\frac{x\sqrt{x^2+a^2}}{2}+\frac{a^2}{2}\ln\left(x+\sqrt{x^2+a^2}\right) +$$ +<<*>>= +)clear all + +--S 8 of 28 +aa:=integrate(sqrt(x^2+a^2),x) +--R +--R +--R (1) +--R +-------+ +-------+ +--R 2 | 2 2 2 2 4 | 2 2 +--R (- 2a x\|x + a + 2a x + a )log(\|x + a - x) +--R + +--R +-------+ +--R 3 2 | 2 2 4 2 2 +--R (- 2x - a x)\|x + a + 2x + 2a x +--R / +--R +-------+ +--R | 2 2 2 2 +--R 4x\|x + a - 4x - 2a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.190~~~~~$\displaystyle\int{x\sqrt{x^2+a^2}}~dx$} +$$\int{x\sqrt{x^2+a^2}}= +\frac{(x^2+a^2)^{3/2}}{3} +$$ +<<*>>= +)clear all + +--S 9 of 28 +aa:=integrate(x*sqrt(x^2+a^2),x) +--R +--R +--R +-------+ +--R 5 2 3 4 | 2 2 6 2 4 4 2 6 +--R (- 4x - 7a x - 3a x)\|x + a + 4x + 9a x + 6a x + a +--R (1) ----------------------------------------------------------- +--R +-------+ +--R 2 2 | 2 2 3 2 +--R (12x + 3a )\|x + a - 12x - 9a x +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.191~~~~~$\displaystyle +\int{x^2\sqrt{x^2+a^2}}~dx$} +$$\int{x^2\sqrt{x^2+a^2}}= +\frac{x(x^2+a^2)^{3/2}}{4}-\frac{a^2x\sqrt{x^2+a^2}}{8}- +\frac{a^4}{8}\ln\left(x+\sqrt{x^2+a^2}\right) +$$ +<<*>>= +)clear all + +--S 10 of 28 +aa:=integrate(x^2*sqrt(x^2+a^2),x) +--R +--R +--R (1) +--R +-------+ +-------+ +--R 4 3 6 | 2 2 4 4 6 2 8 | 2 2 +--R ((8a x + 4a x)\|x + a - 8a x - 8a x - a )log(\|x + a - x) +--R + +--R +-------+ +--R 7 2 5 4 3 6 | 2 2 8 2 6 4 4 6 2 +--R (- 16x - 24a x - 10a x - a x)\|x + a + 16x + 32a x + 20a x + 4a x +--R / +--R +-------+ +--R 3 2 | 2 2 4 2 2 4 +--R (64x + 32a x)\|x + a - 64x - 64a x - 8a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.192~~~~~$\displaystyle +\int{x^3\sqrt{x^2+a^2}}~dx$} +$$\int{x^3\sqrt{x^2+a^2}}= +\frac{(x^2+a^2)^{5/2}}{5}-\frac{a^2(x^2+a^2)^{3/2}}{3} +$$ +<<*>>= +)clear all + +--S 11 of 28 +aa:=integrate(x^3*sqrt(x^2+a^2),x) +--R +--R +--R (1) +--R +-------+ +--R 9 2 7 4 5 6 3 8 | 2 2 10 2 8 +--R (- 48x - 76a x - 3a x + 35a x + 10a x)\|x + a + 48x + 100a x +--R + +--R 4 6 6 4 8 2 10 +--R 35a x - 40a x - 25a x - 2a +--R / +--R +-------+ +--R 4 2 2 4 | 2 2 5 2 3 4 +--R (240x + 180a x + 15a )\|x + a - 240x - 300a x - 75a x +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.193~~~~~$\displaystyle +\int{\frac{\sqrt{x^2+a^2}}{x}}~dx$} +$$\int{\frac{\sqrt{x^2+a^2}}{x}}= +\sqrt{x^2+a^2}-a\ln\left(\frac{a+\sqrt{x^2+a^2}}{x}\right) +$$ +<<*>>= +)clear all + +--S 12 of 28 +aa:=integrate(sqrt(x^2+a^2)/x,x) +--R +--R +--R (1) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R (- a\|x + a + a x)log(\|x + a - x + a) +--R + +--R +-------+ +-------+ +-------+ +--R | 2 2 | 2 2 | 2 2 2 2 +--R (a\|x + a - a x)log(\|x + a - x - a) - x\|x + a + x + a +--R / +--R +-------+ +--R | 2 2 +--R \|x + a - x +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.194~~~~~$\displaystyle +\int{\frac{\sqrt{x^2+a^2}}{x^2}}~dx$} +$$\int{\frac{\sqrt{x^2+a^2}}{x^2}}= +-\frac{\sqrt{x^2+a^2}}{x}+\ln\left(x+\sqrt{x^2+a^2}\right) +$$ +<<*>>= +)clear all + +--S 13 of 28 +aa:=integrate(sqrt(x^2+a^2)/x^2,x) +--R +--R +--R +-------+ +-------+ +--R | 2 2 2 | 2 2 2 +--R (- x\|x + a + x )log(\|x + a - x) - a +--R (1) -------------------------------------------- +--R +-------+ +--R | 2 2 2 +--R x\|x + a - x +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.195~~~~~$\displaystyle +\int{\frac{\sqrt{x^2+a^2}}{x^3}}~dx$} +$$\int{\frac{\sqrt{x^2+a^2}}{x^3}}= +-\frac{\sqrt{x^2+a^2}}{2x^2}-\frac{1}{2a} +\ln\left(\frac{a+\sqrt{x^2+a^2}}{x}\right) +$$ +<<*>>= +)clear all + +--S 14 of 28 +aa:=integrate(sqrt(x^2+a^2)/x^3,x) +--R +--R +--R (1) +--R +-------+ +-------+ +--R 3 | 2 2 4 2 2 | 2 2 +--R (- 2x \|x + a + 2x + a x )log(\|x + a - x + a) +--R + +--R +-------+ +-------+ +--R 3 | 2 2 4 2 2 | 2 2 +--R (2x \|x + a - 2x - a x )log(\|x + a - x - a) +--R + +--R +-------+ +--R 2 3 | 2 2 3 3 +--R (2a x + a )\|x + a - 2a x - 2a x +--R / +--R +-------+ +--R 3 | 2 2 4 3 2 +--R 4a x \|x + a - 4a x - 2a x +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.196~~~~~$\displaystyle\int{\frac{dx}{(x^2+a^2)^{3/2}}}$} +$$\int{\frac{1}{(x^2+a^2)^{3/2}}}= +\frac{x}{a^2\sqrt{x^2+a^2}} +$$ +<<*>>= +)clear all + +--S 15 of 28 +aa:=integrate(1/(x^2+a^2)^(3/2),x) +--R +--R +--R 1 +--R (1) - --------------------- +--R +-------+ +--R | 2 2 2 2 +--R x\|x + a - x - a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.197~~~~~$\displaystyle +\int{\frac{x~dx}{(x^2+a^2)^{3/2}}}$} +$$\int{\frac{x}{(x^2+a^2)^{3/2}}}= +\frac{-1}{\sqrt{x^2+a^2}} +$$ +<<*>>= +)clear all + +--S 16 of 28 +aa:=integrate(x/(x^2+a^2)^(3/2),x) +--R +--R +--R +-------+ +--R | 2 2 +--R \|x + a - x +--R (1) --------------------- +--R +-------+ +--R | 2 2 2 2 +--R x\|x + a - x - a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.198~~~~~$\displaystyle +\int{\frac{x^2dx}{(x^2+a^2)^{3/2}}}$} +$$\int{\frac{x^2}{(x^2+a^2)^{3/2}}}= +\frac{-x}{\sqrt{x^2+a^2}}+\ln\left(x+\sqrt{x^2+a^2}\right) +$$ +<<*>>= +)clear all + +--S 17 of 28 +aa:=integrate(x^2/(x^2+a^2)^(3/2),x) +--R +--R +--R +-------+ +-------+ +--R | 2 2 2 2 | 2 2 2 +--R (- x\|x + a + x + a )log(\|x + a - x) + a +--R (1) ------------------------------------------------- +--R +-------+ +--R | 2 2 2 2 +--R x\|x + a - x - a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.199~~~~~$\displaystyle +\int{\frac{x^3dx}{(x^2+a^2)^{3/2}}}$} +$$\int{\frac{x^3}{(x^2+a^2)^{3/2}}}= +\sqrt{x^2+a^2}+\frac{a^2}{\sqrt{x^2+a^2}} +$$ +<<*>>= +)clear all + +--S 18 of 28 +aa:=integrate(x^3/(x^2+a^2)^(3/2),x) +--R +--R +--R +-------+ +--R 3 2 | 2 2 4 2 2 4 +--R (- 2x - 4a x)\|x + a + 2x + 5a x + 2a +--R (1) -------------------------------------------- +--R +-------+ +--R 2 2 | 2 2 3 2 +--R (2x + a )\|x + a - 2x - 2a x +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.200~~~~~$\displaystyle +\int{\frac{dx}{x(x^2+a^2)^{3/2}}}$} +$$\int{\frac{1}{x(x^2+a^2)^{3/2}}}= +\frac{1}{a^2\sqrt{x^2+a^2}}- +\frac{1}{a^3}\ln\left(\frac{a+\sqrt{x^2+a^2}}{x}\right) +$$ +<<*>>= +)clear all + +--S 19 of 28 +aa:=integrate(1/(x*(x^2+a^2)^(3/2)),x) +--R +--R +--R (1) +--R +-------+ +-------+ +--R | 2 2 2 2 | 2 2 +--R (- x\|x + a + x + a )log(\|x + a - x + a) +--R + +--R +-------+ +-------+ +-------+ +--R | 2 2 2 2 | 2 2 | 2 2 +--R (x\|x + a - x - a )log(\|x + a - x - a) - a\|x + a + a x +--R / +--R +-------+ +--R 3 | 2 2 3 2 5 +--R a x\|x + a - a x - a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.201~~~~~$\displaystyle +\int{\frac{dx}{x^2(x^2+a^2)^{3/2}}}$} +$$\int{\frac{1}{x^2(x^2+a^2)^{3/2}}}= +-\frac{\sqrt{x^2+a^2}}{a^4x}-\frac{x}{a^4\sqrt{x^2+a^2}} +$$ +<<*>>= +)clear all + +--S 20 of 28 +aa:=integrate(1/(x^2*(x^2+a^2)^(3/2)),x) +--R +--R +--R 1 +--R (1) - ----------------------------------- +--R +-------+ +--R 3 2 | 2 2 4 2 2 +--R (2x + a x)\|x + a - 2x - 2a x +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.202~~~~~$\displaystyle +\int{\frac{dx}{x^3(x^2+a^2)^{3/2}}}$} +$$\int{\frac{1}{x^3(x^2+a^2)^{3/2}}}= +\frac{-1}{2a^2x^2\sqrt{x^2+a^2}}- +\frac{3}{2a^4\sqrt{x^2+a^2}}+ +\frac{3}{2a^5}\ln\left(\frac{a+\sqrt{x^2+a^2}}{x}\right) +$$ +<<*>>= +)clear all + +--S 21 of 28 +aa:=integrate(1/(x^3*(x^2+a^2)^(3/2)),x) +--R +--R +--R (1) +--R +-------+ +-------+ +--R 5 2 3 | 2 2 6 2 4 4 2 | 2 2 +--R ((12x + 9a x )\|x + a - 12x - 15a x - 3a x )log(\|x + a - x + a) +--R + +--R +-------+ +--R 5 2 3 | 2 2 6 2 4 4 2 +--R ((- 12x - 9a x )\|x + a + 12x + 15a x + 3a x ) +--R * +--R +-------+ +--R | 2 2 +--R log(\|x + a - x - a) +--R + +--R +-------+ +--R 4 3 2 5 | 2 2 5 3 3 5 +--R (12a x + 7a x + a )\|x + a - 12a x - 13a x - 3a x +--R / +--R +-------+ +--R 5 5 7 3 | 2 2 5 6 7 4 9 2 +--R (8a x + 6a x )\|x + a - 8a x - 10a x - 2a x +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.203~~~~~$\displaystyle\int{(x^2+a^2)^{3/2}}~dx$} +$$\int{(x^2+a^2)^{3/2}}= +\frac{x(x^2+a^2)^{3/2}}{4}+\frac{3a^2x\sqrt{x^2+a^2}}{8}+ +\frac{3}{8}a^4\ln\left(x+\sqrt{x^2+a^2}\right) +$$ +<<*>>= +)clear all + +--S 22 of 28 +aa:=integrate((x^2+a^2)^(3/2),x) +--R +--R +--R (1) +--R +-------+ +-------+ +--R 4 3 6 | 2 2 4 4 6 2 8 | 2 2 +--R ((- 24a x - 12a x)\|x + a + 24a x + 24a x + 3a )log(\|x + a - x) +--R + +--R +-------+ +--R 7 2 5 4 3 6 | 2 2 8 2 6 4 4 +--R (- 16x - 56a x - 42a x - 5a x)\|x + a + 16x + 64a x + 68a x +--R + +--R 6 2 +--R 20a x +--R / +--R +-------+ +--R 3 2 | 2 2 4 2 2 4 +--R (64x + 32a x)\|x + a - 64x - 64a x - 8a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.204~~~~~$\displaystyle\int{x(x^2+a^2)^{3/2}}~dx$} +$$\int{x(x^2+a^2)^{3/2}}=\frac{(x^2+a^2)^{5/2}}{5}$$ +<<*>>= +)clear all + +--S 23 of 28 +aa:=integrate(x*(x^2+a^2)^(3/2),x) +--R +--R +--R (1) +--R +-------+ +--R 9 2 7 4 5 6 3 8 | 2 2 10 2 8 +--R (- 16x - 52a x - 61a x - 30a x - 5a x)\|x + a + 16x + 60a x +--R + +--R 4 6 6 4 8 2 10 +--R 85a x + 55a x + 15a x + a +--R / +--R +-------+ +--R 4 2 2 4 | 2 2 5 2 3 4 +--R (80x + 60a x + 5a )\|x + a - 80x - 100a x - 25a x +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.205~~~~~$\displaystyle\int{x^2(x^2+a^2)^{3/2}}~dx$} +$$\int{x^2(x^2+a^2)^{3/2}}= +\frac{x(x^2+a^2)^{5/2}}{6}-\frac{a^2x(x^2+a^2)^{3/2}}{24}- +\frac{a^4x\sqrt{x^2+a^2}}{16}- +\frac{a^6}{16}\ln\left(x+\sqrt{x^2+a^2}\right) +$$ +<<*>>= +)clear all + +--S 24 of 28 +aa:=integrate(x^2*(x^2+a^2)^(3/2),x) +--R +--R +--R (1) +--R +-------+ +--R 6 5 8 3 10 | 2 2 6 6 8 4 10 2 +--R (96a x + 96a x + 18a x)\|x + a - 96a x - 144a x - 54a x +--R + +--R 12 +--R - 3a +--R * +--R +-------+ +--R | 2 2 +--R log(\|x + a - x) +--R + +--R +-------+ +--R 11 2 9 4 7 6 5 8 3 10 | 2 2 +--R (- 256x - 832a x - 912a x - 404a x - 68a x - 3a x)\|x + a +--R + +--R 12 2 10 4 8 6 6 8 4 10 2 +--R 256x + 960a x + 1296a x + 772a x + 198a x + 18a x +--R / +--R +-------+ +--R 5 2 3 4 | 2 2 6 2 4 4 2 6 +--R (1536x + 1536a x + 288a x)\|x + a - 1536x - 2304a x - 864a x - 48a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.206~~~~~$\displaystyle\int{x^3(x^2+a^2)^{3/2}}~dx$} +$$\int{x^3(x^2+a^2)^{3/2}}= +\frac{(x^2+a^2)^{7/2}}{7}-\frac{a^2(x^2+a^2)^{5/2}}{5} +$$ +<<*>>= +)clear all + +--S 25 of 28 +aa:=integrate(x^3*(x^2+a^2)^(3/2),x) +--R +--R +--R (1) +--R 13 2 11 4 9 6 7 8 5 10 3 +--R - 320x - 1072a x - 1240a x - 467a x + 112a x + 105a x +--R + +--R 12 +--R 14a x +--R * +--R +-------+ +--R | 2 2 +--R \|x + a +--R + +--R 14 2 12 4 10 6 8 8 6 10 4 12 2 +--R 320x + 1232a x + 1736a x + 973a x + 21a x - 175a x - 49a x +--R + +--R 14 +--R - 2a +--R / +--R +-------+ +--R 6 2 4 4 2 6 | 2 2 7 2 5 +--R (2240x + 2800a x + 840a x + 35a )\|x + a - 2240x - 3920a x +--R + +--R 4 3 6 +--R - 1960a x - 245a x +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.207~~~~~$\displaystyle +\int{\frac{(x^2+a^2)^{3/2}}{x}}~dx$} +$$\int{\frac{(x^2+a^2)^{3/2}}{x}}= +\frac{(x^2+a^2)^{3/2}}{3}+a^2\sqrt{x^2+a^2}- +a^3\ln\left(\frac{a+\sqrt{x^2+a^2}}{x}\right) +$$ +<<*>>= +)clear all + +--S 26 of 28 +aa:=integrate((x^2+a^2)^(3/2)/x,x) +--R +--R +--R (1) +--R +-------+ +-------+ +--R 3 2 5 | 2 2 3 3 5 | 2 2 +--R ((- 12a x - 3a )\|x + a + 12a x + 9a x)log(\|x + a - x + a) +--R + +--R +-------+ +-------+ +--R 3 2 5 | 2 2 3 3 5 | 2 2 +--R ((12a x + 3a )\|x + a - 12a x - 9a x)log(\|x + a - x - a) +--R + +--R +-------+ +--R 5 2 3 4 | 2 2 6 2 4 4 2 6 +--R (- 4x - 19a x - 12a x)\|x + a + 4x + 21a x + 21a x + 4a +--R / +--R +-------+ +--R 2 2 | 2 2 3 2 +--R (12x + 3a )\|x + a - 12x - 9a x +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.208~~~~~$\displaystyle +\int{\frac{(x^2+a^2)^{3/2}}{x^2}}~dx$} +$$\int{\frac{(x^2+a^2)^{3/2}}{x^2}}= +-\frac{(x^2+a^2)^{3/2}}{x}+\frac{3x\sqrt{x^2+a^2}}{2}+ +\frac{3}{2}a^2\ln\left(x+\sqrt{x^2+a^2}\right) +$$ +<<*>>= +)clear all + +--S 27 of 28 +aa:=integrate((x^2+a^2)^{3/2}/x^2,x) +--R +--R +--R (1) +--R +-------+ +-------+ +--R 2 3 4 | 2 2 2 4 4 2 | 2 2 +--R ((- 12a x - 3a x)\|x + a + 12a x + 9a x )log(\|x + a - x) +--R + +--R +-------+ +--R 5 2 3 4 | 2 2 6 2 4 4 2 6 +--R (- 4x - 3a x + 4a x)\|x + a + 4x + 5a x - 3a x - 2a +--R / +--R +-------+ +--R 3 2 | 2 2 4 2 2 +--R (8x + 2a x)\|x + a - 8x - 6a x +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.209~~~~~$\displaystyle +\int{\frac{(x^2+a^2)^{3/2}}{x^3}}~dx$} +$$\int{\frac{(x^2+a^2)^{3/2}}{x^3}}= +-\frac{(x^2+a^2)^{3/2}}{2x^2}+\frac{3}{2}\sqrt{x^2+a^2}- +\frac{3}{2}a\ln\left(\frac{a+\sqrt{x^2+a^2}}{x}\right) +$$ +<<*>>= +)clear all + +--S 28 of 28 +aa:=integrate((x^2+a^2)^(3/2)/x^3,x) +--R +--R +--R (1) +--R +-------+ +-------+ +--R 4 3 2 | 2 2 5 3 3 | 2 2 +--R ((- 12a x - 3a x )\|x + a + 12a x + 9a x )log(\|x + a - x + a) +--R + +--R +-------+ +-------+ +--R 4 3 2 | 2 2 5 3 3 | 2 2 +--R ((12a x + 3a x )\|x + a - 12a x - 9a x )log(\|x + a - x - a) +--R + +--R +-------+ +--R 5 2 3 4 | 2 2 6 2 4 4 2 6 +--R (- 8x - 2a x + 3a x)\|x + a + 8x + 6a x - 3a x - a +--R / +--R +-------+ +--R 4 2 2 | 2 2 5 2 3 +--R (8x + 2a x )\|x + a - 8x - 6a x +--R Type: Union(Expression Integer,...) +--E + +)spool +)lisp (bye) +@ + +\eject +\begin{thebibliography}{99} +\bibitem{1} Spiegel, Murray R. +{\sl Mathematical Handbook of Formulas and Tables}\\ +Schaum's Outline Series McGraw-Hill 1968 p67-68 +\end{thebibliography} +\end{document}