diff --git a/changelog b/changelog index 6b98774..a9a8d1b 100644 --- a/changelog +++ b/changelog @@ -1,3 +1,5 @@ +20080331 tpd src/input/Makefile add integration regression testing +20080331 tpd src/input/schaum12.input integrals of ax^2+bx+c 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) diff --git a/src/input/Makefile.pamphlet b/src/input/Makefile.pamphlet index a1382e4..e42c5c0 100644 --- a/src/input/Makefile.pamphlet +++ b/src/input/Makefile.pamphlet @@ -357,7 +357,7 @@ REGRES= algaggr.regress algbrbf.regress algfacob.regress alist.regress \ roman.regress roots.regress ruleset.regress rules.regress \ schaum1.regress schaum2.regress schaum3.regress schaum4.regress \ schaum5.regress schaum6.regress schaum7.regress schaum8.regress \ - schaum9.regress schaum10.regress schaum11.regress \ + schaum9.regress schaum10.regress schaum11.regress schaum12.regress \ scherk.regress scope.regress seccsc.regress \ segbind.regress seg.regress \ series2.regress series.regress sersolve.regress set.regress \ @@ -633,7 +633,7 @@ FILES= ${OUT}/algaggr.input ${OUT}/algbrbf.input ${OUT}/algfacob.input \ ${OUT}/schaum2.input ${OUT}/schaum3.input ${OUT}/schaum4.input \ ${OUT}/schaum5.input ${OUT}/schaum6.input ${OUT}/schaum7.input \ ${OUT}/schaum8.input ${OUT}/schaum9.input ${OUT}/schaum10.input \ - ${OUT}/schaum11.input \ + ${OUT}/schaum11.input ${OUT}/schaum12.input \ ${OUT}/saddle.input \ ${OUT}/scherk.input ${OUT}/scope.input ${OUT}/seccsc.input \ ${OUT}/segbind.input ${OUT}/seg.input ${OUT}/series2.input \ @@ -937,7 +937,7 @@ DOCFILES= \ ${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}/schaum11.input.dvi ${DOC}/schaum12.input.dvi \ ${DOC}/s01eaf.input.dvi ${DOC}/s13aaf.input.dvi \ ${DOC}/s13acf.input.dvi ${DOC}/s13adf.input.dvi \ ${DOC}/s14aaf.input.dvi ${DOC}/s14abf.input.dvi \ diff --git a/src/input/schaum12.input.pamphlet b/src/input/schaum12.input.pamphlet new file mode 100644 index 0000000..6027801 --- /dev/null +++ b/src/input/schaum12.input.pamphlet @@ -0,0 +1,780 @@ +\documentclass{article} +\usepackage{axiom} +\begin{document} +\title{\$SPAD/input schaum12.input} +\author{Timothy Daly} +\maketitle +\eject +\tableofcontents +\eject +\section{\cite{1}:14.265~~~~~$\displaystyle +\int{\frac{dx}{ax^2+bx+c}}$} +$$\int{\frac{1}{ax^2+bx+c}}= +\left\{ +\begin{array}{l} +\displaystyle\frac{2}{\sqrt{4ac-b^2}}\tan^{-1}\frac{2ax+b}{\sqrt{4ac-b^2}}\\ +\displaystyle\frac{1}{\sqrt{b^2-4ac}}\ln\left( +\frac{2ax+b-\sqrt{b^2-4ac}}{2ax+b+\sqrt{b^2-4ac}}\right) +\end{array} +\right. +$$ +<<*>>= +)spool schaum12.output +)set message test on +)set message auto off +)clear all + +--S 1 of 15 +aa:=integrate(1/sqrt(a*x^2+b*x+c),x) +--R +--R +--R (1) +--R [ +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R / +--R +-+ +--R \|a +--R , +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R \|- a \|a x + b x + c - \|- a \|c +--R 2atan(------------------------------------) +--R a x +--R -------------------------------------------] +--R +---+ +--R \|- a +--R Type: Union(List Expression Integer,...) +--E +@ + +\section{\cite{1}:14.266~~~~~$\displaystyle +\int{\frac{x~dx}{ax^2+bx+c}}$} +$$\int{\frac{x}{ax^2+bx+c}}= +\frac{1}{2a}\ln(ax^2+bx+c)-\frac{b}{2a}\int{\frac{1}{ax^2+bx+c}} +$$ +<<*>>= +)clear all + +--S 2 of 15 +aa:=integrate(x/sqrt(a*x^2+b*x+c),x) +--R +--R +--R (1) +--R [ +--R +--------------+ +--R +-+ | 2 2 +--R (2b\|c \|a x + b x + c - b x - 2b c) +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c + 2a x)\|a x + b x + c - 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R +--------------+ +--R +-+ | 2 2 +-+ +-+ +--R - 2b x\|a \|a x + b x + c + (4a x + 2b x)\|a \|c +--R / +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R 4a\|a \|c \|a x + b x + c + (- 2a b x - 4a c)\|a +--R , +--R +--R +--------------+ +--R +-+ | 2 2 +--R (- 2b\|c \|a x + b x + c + b x + 2b c) +--R * +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R \|- a \|a x + b x + c - \|- a \|c +--R atan(------------------------------------) +--R a x +--R + +--R +--------------+ +--R +---+ | 2 2 +---+ +-+ +--R - b x\|- a \|a x + b x + c + (2a x + b x)\|- a \|c +--R / +--R +--------------+ +--R +---+ +-+ | 2 +---+ +--R 2a\|- a \|c \|a x + b x + c + (- a b x - 2a c)\|- a +--R ] +--R Type: Union(List Expression Integer,...) +--E +@ + +\section{\cite{1}:14.267~~~~~$\displaystyle +\int{\frac{x^2dx}{ax^2+bx+c}}$} +$$\int{\frac{x^2}{ax^2+bx+c}}= +\frac{x}{a}-\frac{b}{2a^2}\ln(ax^2+bx+c)+\frac{b^2-2ac}{2a^2} +\int{\frac{1}{ax^2+bx+c}} +$$ +<<*>>= +)clear all + +--S 3 of 15 +aa:=integrate(x^2/(a*x^2+b*x+c),x) +--R +--R +--R (1) +--R [ +--R 2 +--R (2a c - b ) +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (- 8a c + 2a b )x +--R + +--R 3 +--R - 4a b c + b +--R / +--R 2 +--R a x + b x + c +--R + +--R +-----------+ +--R 2 | 2 +--R (- b log(a x + b x + c) + 2a x)\|- 4a c + b +--R / +--R +-----------+ +--R 2 | 2 +--R 2a \|- 4a c + b +--R , +--R +--R +---------+ +--R | 2 +--R 2 (2a x + b)\|4a c - b +--R (- 4a c + 2b )atan(----------------------) +--R 2 +--R 4a c - b +--R + +--R +---------+ +--R 2 | 2 +--R (- b log(a x + b x + c) + 2a x)\|4a c - b +--R / +--R +---------+ +--R 2 | 2 +--R 2a \|4a c - b +--R ] +--R Type: Union(List Expression Integer,...) +--E +@ + +\section{\cite{1}:14.268~~~~~$\displaystyle +\int{\frac{x^m~dx}{ax^2+bx+c}}$} +$$\int{\frac{x^m}{ax^2+bx+c}}= +\frac{x^{m-1}}{(m-1)a}-\frac{c}{a}\int{\frac{x^{m-2}}{ax^2+bx+c}}- +\frac{b}{a}\int{\frac{x^{m-1}}{ax^2+bx+c}} +$$ +<<*>>= +)clear all + +--S 4 of 15 +aa:=integrate(x^m/(a*x^2+b*x+c),x) +--R +--R +--R x m +--R ++ %N +--R (1) | --------------- d%N +--R ++ 2 +--R c + %N b + %N a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.269~~~~~$\displaystyle +\int{\frac{dx}{x(ax^2+bx+c)}}$} +$$\int{\frac{1}{x(ax^2+bx+c)}}= +\frac{1}{2c}\ln\left(\frac{x^2}{ax^2+bx+c}\right)- +\frac{b}{2c}\int{\frac{1}{ax^2+bx+c}} +$$ +<<*>>= +)clear all + +--S 5 of 15 +aa:=integrate(1/(x*(a*x^2+b*x+c)),x) +--R +--R +--R (1) +--R [ +--R b +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (- 8a c + 2a b )x +--R + +--R 3 +--R - 4a b c + b +--R / +--R 2 +--R a x + b x + c +--R + +--R +-----------+ +--R 2 | 2 +--R (- log(a x + b x + c) + 2log(x))\|- 4a c + b +--R / +--R +-----------+ +--R | 2 +--R 2c\|- 4a c + b +--R , +--R +--R +---------+ +--R | 2 +--R (2a x + b)\|4a c - b +--R - 2b atan(----------------------) +--R 2 +--R 4a c - b +--R + +--R +---------+ +--R 2 | 2 +--R (- log(a x + b x + c) + 2log(x))\|4a c - b +--R / +--R +---------+ +--R | 2 +--R 2c\|4a c - b +--R ] +--R Type: Union(List Expression Integer,...) +--E +@ + +\section{\cite{1}:14.270~~~~~$\displaystyle +\int{\frac{dx}{x^2(ax^2+bx+c)}}$} +$$\int{\frac{1}{x^2(ax^2+bx+c)}}= +\frac{b}{2c^2}\ln\left(\frac{ax^2+bx+c}{x^2}\right)-\frac{1}{cx}+ +\frac{b^2-2ac}{2c^2}\int{\frac{1}{ax^2+bx+c}} +$$ +<<*>>= +)clear all + +--S 6 of 15 +aa:=integrate(1/(x^2*(a*x^2+b*x+c)),x) +--R +--R +--R (1) +--R [ +--R 2 +--R (2a c - b )x +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (- 8a c + 2a b )x +--R + +--R 3 +--R - 4a b c + b +--R / +--R 2 +--R a x + b x + c +--R + +--R +-----------+ +--R 2 | 2 +--R (b x log(a x + b x + c) - 2b x log(x) - 2c)\|- 4a c + b +--R / +--R +-----------+ +--R 2 | 2 +--R 2c x\|- 4a c + b +--R , +--R +--R +---------+ +--R | 2 +--R 2 (2a x + b)\|4a c - b +--R (- 4a c + 2b )x atan(----------------------) +--R 2 +--R 4a c - b +--R + +--R +---------+ +--R 2 | 2 +--R (b x log(a x + b x + c) - 2b x log(x) - 2c)\|4a c - b +--R / +--R +---------+ +--R 2 | 2 +--R 2c x\|4a c - b +--R ] +--R Type: Union(List Expression Integer,...) +--E +@ + +\section{\cite{1}:14.271~~~~~$\displaystyle +\int{\frac{dx}{x^n(ax^2+bx+c)}}$} +$$\int{\frac{1}{x^n(ax^2+bx+c)}}= +-\frac{1}{(n-1)cx^{n-1}}- +\frac{b}{c}\int{\frac{1}{x^{n-1}(ax^2+bx+c)}}- +\frac{a}{c}\int{\frac{1}{x^{x-2}(ax^2+bx+c)}} +$$ +<<*>>= +)clear all + +--S 7 of 15 +aa:=integrate(1/(x^n*(ax^2+bx+c)),x) +--R +--R +--R x +--R (1) - ------------------------------------------ +--R 2 2 n log(x) +--R ((c + bx + ax )n - c - bx - ax )%e +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.272~~~~~$\displaystyle +\int{\frac{dx}{(ax^2+bx+c)^2}}$} +$$\int{\frac{1}{(ax^2+bx+c)^2}}= +\frac{2xa+b}{(4ac-b^2)(ax^2+bx+c)}+ +\frac{2a}{4ac-b^2}\int{\frac{1}{ax^2+bx+c}} +$$ +<<*>>= +)clear all + +--S 8 of 15 +aa:=integrate(1/(a*x^2+b*x+c)^2,x) +--R +--R +--R (1) +--R [ +--R 2 2 +--R (2a x + 2a b x + 2a c) +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x +--R + +--R 3 +--R 4a b c - b +--R / +--R 2 +--R a x + b x + c +--R + +--R +-----------+ +--R | 2 +--R (2a x + b)\|- 4a c + b +--R / +--R +-----------+ +--R 2 2 2 3 2 2 | 2 +--R ((4a c - a b )x + (4a b c - b )x + 4a c - b c)\|- 4a c + b +--R , +--R +---------+ +--R | 2 +---------+ +--R 2 2 (2a x + b)\|4a c - b | 2 +--R (4a x + 4a b x + 4a c)atan(----------------------) + (2a x + b)\|4a c - b +--R 2 +--R 4a c - b +--R ---------------------------------------------------------------------------- +--R +---------+ +--R 2 2 2 3 2 2 | 2 +--R ((4a c - a b )x + (4a b c - b )x + 4a c - b c)\|4a c - b +--R ] +--R Type: Union(List Expression Integer,...) +--E +@ + +\section{\cite{1}:14.273~~~~~$\displaystyle +\int{\frac{x~dx}{(ax^2+bx+c)^2}}$} +$$\int{\frac{x}{(ax^2+bx+c)^2}}= +-\frac{bx+2c}{(4ac-b^2)(ax^2+bx+c)}- +\frac{b}{4ac-b^2}\int{\frac{1}{ax^2+bx+c}} +$$ +<<*>>= +)clear all + +--S 9 of 15 +aa:=integrate(x/(a*x^2+b*x+c)^2,x) +--R +--R +--R (1) +--R [ +--R 2 2 +--R (a b x + b x + b c) +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (- 8a c + 2a b )x +--R + +--R 3 +--R - 4a b c + b +--R / +--R 2 +--R a x + b x + c +--R + +--R +-----------+ +--R | 2 +--R (- b x - 2c)\|- 4a c + b +--R / +--R +-----------+ +--R 2 2 2 3 2 2 | 2 +--R ((4a c - a b )x + (4a b c - b )x + 4a c - b c)\|- 4a c + b +--R , +--R +--R +---------+ +--R | 2 +--R 2 2 (2a x + b)\|4a c - b +--R (- 2a b x - 2b x - 2b c)atan(----------------------) +--R 2 +--R 4a c - b +--R + +--R +---------+ +--R | 2 +--R (- b x - 2c)\|4a c - b +--R / +--R +---------+ +--R 2 2 2 3 2 2 | 2 +--R ((4a c - a b )x + (4a b c - b )x + 4a c - b c)\|4a c - b +--R ] +--R Type: Union(List Expression Integer,...) +--E +@ + +\section{\cite{1}:14.274~~~~~$\displaystyle +\int{\frac{x^2~dx}{(ax^2+bx+c)^2}}$} +$$\int{\frac{x^2}{(ax^2+bx+c)^2}}= +\frac{(b^2-2ac)x+bc}{a(4ac-b^2)(ax^2+bx+c)}+ +\frac{2c}{4ac-b^2}\int{\frac{1}{ax^2+bx+c}} +$$ +<<*>>= +)clear all + +--S 10 of 15 +aa:=integrate(x^2/(a*x^2+b*x+c)^2,x) +--R +--R +--R (1) +--R [ +--R 2 2 2 +--R (2a c x + 2a b c x + 2a c ) +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x +--R + +--R 3 +--R 4a b c - b +--R / +--R 2 +--R a x + b x + c +--R + +--R +-----------+ +--R 2 | 2 +--R ((- 2a c + b )x + b c)\|- 4a c + b +--R / +--R +-----------+ +--R 3 2 2 2 2 3 2 2 2 | 2 +--R ((4a c - a b )x + (4a b c - a b )x + 4a c - a b c)\|- 4a c + b +--R , +--R +--R +---------+ +--R | 2 +--R 2 2 2 (2a x + b)\|4a c - b +--R (4a c x + 4a b c x + 4a c )atan(----------------------) +--R 2 +--R 4a c - b +--R + +--R +---------+ +--R 2 | 2 +--R ((- 2a c + b )x + b c)\|4a c - b +--R / +--R +---------+ +--R 3 2 2 2 2 3 2 2 2 | 2 +--R ((4a c - a b )x + (4a b c - a b )x + 4a c - a b c)\|4a c - b +--R ] +--R Type: Union(List Expression Integer,...) +--E +@ + +\section{\cite{1}:14.275~~~~~$\displaystyle +\int{\frac{x^m~dx}{(ax^2+bx+c)^n}}$} +$$ +\begin{array}{lr} +\displaystyle\int{\frac{x^m}{(ax^2+bx+c)^n}}= +&\displaystyle-\frac{x^{m-1}}{(2n-m-1)a(ax^2+bx+c)^{n-1}}\\ +&\\ +&\displaystyle+\frac{(m-1)c}{(2n-m-1)a}\int{\frac{x^{m-2}}{(ax^2+bx+2)^n}}\\ +&\\ +&\displaystyle-\frac{(n-m)b}{(2n-m-1)a}\int{\frac{x^{m-1}}{(ax^2+bx+c)^n}} +\end{array} +$$ +<<*>>= +)clear all + +--S 11 of 15 +aa:=integrate(x^m/(a*x^2+b*x+c)^n,x) +--R +--R +--R x m +--R ++ %N +--R (1) | ------------------ d%N +--R ++ 2 n +--R (c + %N b + %N a) +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.276~~~~~$\displaystyle +\int{\frac{x^{2n-1}~dx}{(ax^2+bx+c)^n}}$} +$$\begin{array}{lr} +\displaystyle\int{\frac{x^{2n-1}}{(ax^2+bx+c)^n}}= +&\displaystyle\frac{1}{a}\int{\frac{x^{2n-3}}{(ax^2+bx+c)^{n-1}}}\\ +&\\ +&\displaystyle-\frac{c}{a}\int{\frac{x^{2n-3}}{(ax^2+bx+c)^n}}\\ +&\\ +&\displaystyle-\frac{b}{a}\int{\frac{x^{2n-2}}{(ax^2+bx+c)^n}} +\end{array} +$$ +<<*>>= +)clear all + +--S 12 of 15 +aa:=integrate(x^(2*n-1)/(a*x^2+b*x+c)^n,x) +--R +--R +--R x 2n - 1 +--R ++ %N +--R (1) | ------------------ d%N +--R ++ 2 n +--R (c + %N b + %N a) +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.277~~~~~$\displaystyle +\int{\frac{dx}{x(ax^2+bx+c)^2}}$} +$$\begin{array}{lr} +\displaystyle\int{\frac{1}{x(ax^2+bx+c)^2}}= +&\displaystyle\frac{1}{2c(ax^2+bx+2)}\\ +&\\ +&\displaystyle-\frac{b}{2c}\int{\frac{1}{(ax^2+bx+c)^2}}\\ +&\\ +&\displaystyle+\frac{1}{c}\int{\frac{1}{x(ax^2+bx+c)}} +\end{array} +$$ +<<*>>= +)clear all + +--S 13 of 15 +aa:=integrate(1/(x*(a*x^2+b*x+c)^2),x) +--R +--R +--R (1) +--R [ +--R 2 3 2 2 4 2 3 +--R ((6a b c - a b )x + (6a b c - b )x + 6a b c - b c) +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (- 8a c + 2a b )x +--R + +--R 3 +--R - 4a b c + b +--R / +--R 2 +--R a x + b x + c +--R + +--R 2 2 2 3 2 2 +--R ((- 4a c + a b )x + (- 4a b c + b )x - 4a c + b c) +--R * +--R 2 +--R log(a x + b x + c) +--R + +--R 2 2 2 3 2 2 +--R ((8a c - 2a b )x + (8a b c - 2b )x + 8a c - 2b c)log(x) +--R + +--R 2 2 +--R - 2a b c x + 4a c - 2b c +--R * +--R +-----------+ +--R | 2 +--R \|- 4a c + b +--R / +--R +-----------+ +--R 2 3 2 2 2 3 3 2 4 2 3 | 2 +--R ((8a c - 2a b c )x + (8a b c - 2b c )x + 8a c - 2b c )\|- 4a c + b +--R , +--R +--R 2 3 2 2 4 2 3 +--R ((- 12a b c + 2a b )x + (- 12a b c + 2b )x - 12a b c + 2b c) +--R * +--R +---------+ +--R | 2 +--R (2a x + b)\|4a c - b +--R atan(----------------------) +--R 2 +--R 4a c - b +--R + +--R 2 2 2 3 2 2 +--R ((- 4a c + a b )x + (- 4a b c + b )x - 4a c + b c) +--R * +--R 2 +--R log(a x + b x + c) +--R + +--R 2 2 2 3 2 2 +--R ((8a c - 2a b )x + (8a b c - 2b )x + 8a c - 2b c)log(x) +--R + +--R 2 2 +--R - 2a b c x + 4a c - 2b c +--R * +--R +---------+ +--R | 2 +--R \|4a c - b +--R / +--R +---------+ +--R 2 3 2 2 2 3 3 2 4 2 3 | 2 +--R ((8a c - 2a b c )x + (8a b c - 2b c )x + 8a c - 2b c )\|4a c - b +--R ] +--R Type: Union(List Expression Integer,...) +--E +@ + +\section{\cite{1}:14.278~~~~~$\displaystyle +\int{\frac{dx}{x^2(ax^2+bx+c)^2}}$} +$$\begin{array}{lr} +\displaystyle\int{\frac{1}{x^2(ax^2+bx+c)^2}}= +&\displaystyle-\frac{1}{cx(ax^2+bx+c)}\\ +&\\ +&\displaystyle-\frac{3a}{c}\int{\frac{1}{(ax^2+bx+c)^2}}\\ +&\\ +&\displaystyle-\frac{2b}{c}\int{\frac{1}{x(ax^2+bx+c)^2}} +\end{array} +$$ +<<*>>= +)clear all + +--S 14 of 15 +aa:=integrate(1/(x^2*(a*x^2+b*x+c)^2),x) +--R +--R +--R (1) +--R [ +--R 3 2 2 2 4 3 2 2 3 5 2 +--R (6a c - 6a b c + a b )x + (6a b c - 6a b c + b )x +--R + +--R 2 3 2 2 4 +--R (6a c - 6a b c + b c)x +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (- 8a c + 2a b )x +--R + +--R 3 +--R - 4a b c + b +--R / +--R 2 +--R a x + b x + c +--R + +--R 2 3 3 2 4 2 2 3 +--R ((4a b c - a b )x + (4a b c - b )x + (4a b c - b c)x) +--R * +--R 2 +--R log(a x + b x + c) +--R + +--R 2 3 3 2 4 2 2 3 +--R ((- 8a b c + 2a b )x + (- 8a b c + 2b )x + (- 8a b c + 2b c)x) +--R * +--R log(x) +--R + +--R 2 2 2 2 2 3 3 2 2 +--R (- 6a c + 2a b c)x + (- 7a b c + 2b c)x - 4a c + b c +--R * +--R +-----------+ +--R | 2 +--R \|- 4a c + b +--R / +--R +-----------+ +--R 2 4 2 3 3 4 3 3 2 5 2 4 | 2 +--R ((4a c - a b c )x + (4a b c - b c )x + (4a c - b c )x)\|- 4a c + b +--R , +--R +--R 3 2 2 2 4 3 2 2 3 5 2 +--R (- 12a c + 12a b c - 2a b )x + (- 12a b c + 12a b c - 2b )x +--R + +--R 2 3 2 2 4 +--R (- 12a c + 12a b c - 2b c)x +--R * +--R +---------+ +--R | 2 +--R (2a x + b)\|4a c - b +--R atan(----------------------) +--R 2 +--R 4a c - b +--R + +--R 2 3 3 2 4 2 2 3 +--R ((4a b c - a b )x + (4a b c - b )x + (4a b c - b c)x) +--R * +--R 2 +--R log(a x + b x + c) +--R + +--R 2 3 3 2 4 2 2 3 +--R ((- 8a b c + 2a b )x + (- 8a b c + 2b )x + (- 8a b c + 2b c)x) +--R * +--R log(x) +--R + +--R 2 2 2 2 2 3 3 2 2 +--R (- 6a c + 2a b c)x + (- 7a b c + 2b c)x - 4a c + b c +--R * +--R +---------+ +--R | 2 +--R \|4a c - b +--R / +--R +---------+ +--R 2 4 2 3 3 4 3 3 2 5 2 4 | 2 +--R ((4a c - a b c )x + (4a b c - b c )x + (4a c - b c )x)\|4a c - b +--R ] +--R Type: Union(List Expression Integer,...) +--E +@ + +\section{\cite{1}:14.279~~~~~$\displaystyle +\int{\frac{dx}{x^m(ax^2+bx+c)^n}}$} +$$\begin{array}{lr} +\displaystyle\int{\frac{1}{x^m(ax^2+bx+c)^n}}= +&\displaystyle-\frac{1}{(m-1)cx^{m-1}(ax^2+bx+c)^{n-1}}\\ +&\\ +&\displaystyle-\frac{(m+2n-3)a}{(m-1)c}\int{\frac{1}{x^{m-2}(ax^2+bx+c)^n}}\\ +&\\ +&\displaystyle-\frac{(m+n-2)b}{(m-1)c}\int{\frac{1}{x^{m-1}(ax^2+bx+c)^n}} +\end{array} +$$ +<<*>>= +)clear all + +--S 15 of 15 +aa:=integrate(1/(x^m*(a*x^2+b*x+c)^n),x) +--R +--R +--R x +--R ++ 1 +--R (1) | --------------------- d%N +--R ++ m 2 n +--R %N (c + %N b + %N a) +--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 p71 +\end{thebibliography} +\end{document}