diff --git a/changelog b/changelog index a244a50..4916b6a 100644 --- a/changelog +++ b/changelog @@ -1,3 +1,9 @@ +20080409 tpd src/input/Makefile add integration regression testing +20080409 tpd src/input/schaum24.input integrals of inverse trig functions +20080409 tpd src/input/schaum23.input integrals of csc(ax) +20080409 tpd src/input/schaum22.input integrals of sec(ax) +20080409 tpd src/input/schaum21.input integrals of cot(ax) +20080409 tpd src/input/schaum20.input integrals of tan(ax) 20080408 tpd src/input/mapleok.input fix I->%i, reorganize 20080406 tpd src/input/Makefile add integration regression testing 20080406 tpd src/input/schaum19.input integrals of sin(ax) and cos(ax) diff --git a/src/input/Makefile.pamphlet b/src/input/Makefile.pamphlet index 8170a48..c184fbe 100644 --- a/src/input/Makefile.pamphlet +++ b/src/input/Makefile.pamphlet @@ -359,7 +359,8 @@ REGRES= algaggr.regress algbrbf.regress algfacob.regress alist.regress \ schaum5.regress schaum6.regress schaum7.regress schaum8.regress \ schaum9.regress schaum10.regress schaum11.regress schaum12.regress \ schaum13.regress schaum14.regress schaum15.regress schaum16.regress \ - schaum17.regress schaum18.regress schaum19.regress \ + schaum17.regress schaum18.regress schaum19.regress schaum20.regress \ + schaum21.regress schaum22.regress schaum23.regress schaum24.regress \ scherk.regress scope.regress seccsc.regress \ segbind.regress seg.regress \ series2.regress series.regress sersolve.regress set.regress \ @@ -638,6 +639,8 @@ FILES= ${OUT}/algaggr.input ${OUT}/algbrbf.input ${OUT}/algfacob.input \ ${OUT}/schaum11.input ${OUT}/schaum12.input ${OUT}/schaum13.input \ ${OUT}/schaum14.input ${OUT}/schaum15.input ${OUT}/schaum16.input \ ${OUT}/schaum17.input ${OUT}/schaum18.input ${OUT}/schaum19.input \ + ${OUT}/schaum20.input ${OUT}/schaum21.input ${OUT}/schaum22.input \ + ${OUT}/schaum23.input ${OUT}/schaum24.input \ ${OUT}/saddle.input \ ${OUT}/scherk.input ${OUT}/scope.input ${OUT}/seccsc.input \ ${OUT}/segbind.input ${OUT}/seg.input ${OUT}/series2.input \ @@ -945,7 +948,9 @@ DOCFILES= \ ${DOC}/schaum13.input.dvi ${DOC}/schaum14.input.dvi \ ${DOC}/schaum15.input.dvi ${DOC}/schaum16.input.dvi \ ${DOC}/schaum17.input.dvi ${DOC}/schaum18.input.dvi \ - ${DOC}/schaum19.input.dvi \ + ${DOC}/schaum19.input.dvi ${DOC}/schaum20.input.dvi \ + ${DOC}/schaum21.input.dvi ${DOC}/schaum22.input.dvi \ + ${DOC}/schaum23.input.dvi ${DOC}/schaum24.input.dvi \ ${DOC}/s01eaf.input.dvi ${DOC}/s13aaf.input.dvi \ ${DOC}/s13acf.input.dvi ${DOC}/s13adf.input.dvi \ ${DOC}/s14aaf.input.dvi ${DOC}/s14abf.input.dvi \ diff --git a/src/input/schaum20.input.pamphlet b/src/input/schaum20.input.pamphlet new file mode 100644 index 0000000..a25209e --- /dev/null +++ b/src/input/schaum20.input.pamphlet @@ -0,0 +1,249 @@ +\documentclass{article} +\usepackage{axiom} +\begin{document} +\title{\$SPAD/input schaum20.input} +\author{Timothy Daly} +\maketitle +\eject +\tableofcontents +\eject +\section{\cite{1}:14.429~~~~~$\displaystyle +\int{\tan{ax}}~dx$} +$$\int{\tan{ax}}= +-\frac{1}{a}\ln~\cos{ax}= +\frac{1}{a}\ln~\sec{ax} +$$ +<<*>>= +)spool schaum20.output +)set message test on +)set message auto off +)clear all + +--S 1 of 11 +aa:=integrate(tan(a*x),x) +--R +--R +--R 2 +--R log(tan(a x) + 1) +--R (1) ------------------ +--R 2a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.430~~~~~$\displaystyle +\int{\tan^2{ax}}~dx$} +$$\int{\tan^2{ax}}= +\frac{\tan{ax}}{x}-x +$$ +<<*>>= +)clear all + +--S 2 of 11 +aa:=integrate(tan(a*x)^2,x) +--R +--R +--R tan(a x) - a x +--R (1) -------------- +--R a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.431~~~~~$\displaystyle +\int{\tan^3{ax}}~dx$} +$$\int{\tan^3{ax}}= +\frac{\tan^2{ax}}{2a}+\frac{1}{a}\ln~\cos{ax} +$$ +<<*>>= +)clear all + +--S 3 of 11 +aa:=integrate(tan(a*x)^3,x) +--R +--R +--R 2 2 +--R - log(tan(a x) + 1) + tan(a x) +--R (1) -------------------------------- +--R 2a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.432~~~~~$\displaystyle +\int{\tan^n{ax}\sec^2{ax}}~dx$} +$$\int{\tan^n{ax}\sec^2{ax}}= +\frac{\tan^{n+1}{ax}}{(n+1)a} +$$ +<<*>>= +)clear all + +--S 4 of 11 +aa:=integrate(tan(a*x)^n*sec(a*x)^2,x) +--R +--R +--R sin(a x) +--R n log(--------) +--R cos(a x) +--R sin(a x)%e +--R (1) ------------------------- +--R (a n + a)cos(a x) +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.433~~~~~$\displaystyle +\int{\frac{\sec^2{ax}}{\tan{ax}}}~dx$} +$$\int{\frac{\sec^2{ax}}{\tan{ax}}}= +\frac{1}{a}\ln~\tan{ax} +$$ +<<*>>= +)clear all + +--S 5 of 11 +aa:=integrate(sec(a*x)^2/tan(a*x),x) +--R +--R +--R sin(a x) 2cos(a x) +--R log(------------) - log(- ------------) +--R cos(a x) + 1 cos(a x) + 1 +--R (1) --------------------------------------- +--R a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.434~~~~~$\displaystyle +\int{\frac{dx}{\tan{ax}}}~dx$} +$$\int{\frac{1}{\tan{ax}}}= +\frac{1}{a}\ln~\sin{ax} +$$ +<<*>>= +)clear all + +--S 6 of 11 +aa:=integrate(1/tan(a*x),x) +--R +--R +--R 2 +--R - log(tan(a x) + 1) + 2log(tan(a x)) +--R (1) ------------------------------------- +--R 2a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.435~~~~~$\displaystyle +\int{x\tan{ax}}~dx$} +$$\int{x\tan{ax}}= +\frac{1}{a^2}\left\{\frac{(ax)^3}{3}+\frac{(ax)^5}{15}+\frac{2(ax)^7}{105} ++\cdots+\frac{2^{2n}(2^{2n}-1)B_n(ax)^{2n+1}}{(2n+1)!}+\cdots\right\} +$$ +<<*>>= +)clear all + +--S 7 of 11 +aa:=integrate(x*tan(a*x),x) +--R +--R +--R x +--R ++ +--I (1) | %I tan(%I a)d%I +--R ++ +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.436~~~~~$\displaystyle +\int{\frac{\tan{ax}}{x}}~dx$} +$$\int{\frac{\tan{ax}}{x}}= +ax+\frac{(ax)^3}{9}+\frac{2(ax)^5}{75}+\cdots ++\frac{2^{2n}(2^{2n}-1)B_n(ax)^{2n-1}}{(2n-1)(2n)!}+\cdots +$$ +<<*>>= +)clear all + +--S 8 of 11 +aa:=integrate(tan(a*x)/x,x) +--R +--R +--R x +--I ++ tan(%I a) +--I (1) | --------- d%I +--I ++ %I +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.437~~~~~$\displaystyle +\int{x\tan^2{ax}}~dx$} +$$\int{x\tan^2{ax}}= +\frac{x\tan{ax}}{a}+\frac{1}{a^2}\ln~\cos{ax}-\frac{x^2}{2} +$$ +<<*>>= +)clear all + +--S 9 of 11 +aa:=integrate(x*tan(a*x)^2,x) +--R +--R +--R 2 2 2 +--R - log(tan(a x) + 1) + 2a x tan(a x) - a x +--R (1) ------------------------------------------- +--R 2 +--R 2a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.438~~~~~$\displaystyle +\int{\frac{dx}{p+q\tan{ax}}}~dx$} +$$\int{\frac{1}{p+q\tan{ax}}}= +\frac{px}{p^2+q^2}+\frac{q}{a(p^2+q^2)}\ln(q\sin{ax}+p\cos{ax}) +$$ +<<*>>= +)clear all + +--S 10 of 11 +aa:=integrate(1/(p+q*tan(a*x)),x) +--R +--R +--R 2 +--R - q log(tan(a x) + 1) + 2q log(q tan(a x) + p) + 2a p x +--R (1) -------------------------------------------------------- +--R 2 2 +--R 2a q + 2a p +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.439~~~~~$\displaystyle +\int{\tan^n{ax}}~dx$} +$$\int{\tan^n{ax}}= +\frac{\tan^{n-1}{ax}}{(n-1)a}-\int{\tan^{n-2}{ax}} +$$ +<<*>>= +)clear all + +--S 11 of 11 +aa:=integrate(tan(a*x)^n,x) +--R +--R +--R x +--R ++ n +--I (1) | tan(%I a) d%I +--R ++ +--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 p80 +\end{thebibliography} +\end{document} diff --git a/src/input/schaum21.input.pamphlet b/src/input/schaum21.input.pamphlet new file mode 100644 index 0000000..6e02ffe --- /dev/null +++ b/src/input/schaum21.input.pamphlet @@ -0,0 +1,262 @@ +\documentclass{article} +\usepackage{axiom} +\begin{document} +\title{\$SPAD/input schaum21.input} +\author{Timothy Daly} +\maketitle +\eject +\tableofcontents +\eject +\section{\cite{1}:14.440~~~~~$\displaystyle +\int{\cot{ax}}~dx$} +$$\int{\cot{ax}}= +\frac{1}{a}\ln\sin{ax} +$$ +<<*>>= +)spool schaum21.output +)set message test on +)set message auto off +)clear all + +--S 1 of 11 +aa:=integrate(cot(a*x),x) +--R +--R +--R sin(2a x) 2 +--R 2log(-------------) - log(-------------) +--R cos(2a x) + 1 cos(2a x) + 1 +--R (1) ---------------------------------------- +--R 2a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.441~~~~~$\displaystyle +\int{\cot^2{ax}}~dx$} +$$\int{\cot^2{ax}}= +-\frac{\cot{ax}}{a}-x +$$ +<<*>>= +)clear all + +--S 2 of 11 +aa:=integrate(cot(a*x)^2,x) +--R +--R +--R - a x sin(2a x) - cos(2a x) - 1 +--R (1) ------------------------------- +--R a sin(2a x) +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.442~~~~~$\displaystyle +\int{\cot^3{ax}}~dx$} +$$\int{\cot^3{ax}}= +-\frac{\cot^2{ax}}{2a}-\frac{1}{a}\ln\sin{ax} +$$ +<<*>>= +)clear all + +--S 3 of 11 +aa:=integrate(cot(a*x)^3,x) +--R +--R +--R (1) +--R sin(2a x) 2 +--R (- 2cos(2a x) + 2)log(-------------) + (cos(2a x) - 1)log(-------------) +--R cos(2a x) + 1 cos(2a x) + 1 +--R + +--R cos(2a x) + 1 +--R / +--R 2a cos(2a x) - 2a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.443~~~~~$\displaystyle +\int{\cot^n{ax}\csc^2{ax}}~dx$} +$$\int{\cot^n{ax}\csc^2{ax}}= +-\frac{\cot^{n+1}{ax}}{(n+1)a} +$$ +<<*>>= +)clear all + +--S 4 of 11 +aa:=integrate(cot(a*x)^n*csc(a*x)^2,x) +--R +--R +--R cos(a x) +--R n log(--------) +--R sin(a x) +--R cos(a x)%e +--R (1) - ------------------------- +--R (a n + a)sin(a x) +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.444~~~~~$\displaystyle +\int{\frac{\csc^2{ax}}{\cot{ax}}}~dx$} +$$\int{\frac{\csc^2{ax}}{\cot{ax}}}= +-\frac{1}{a}\ln\cot{ax} +$$ +<<*>>= +)clear all + +--S 5 of 11 +aa:=integrate(csc(a*x)^2/cot(a*x),x) +--R +--R +--R sin(a x) 2cos(a x) +--R log(------------) - log(- ------------) +--R cos(a x) + 1 cos(a x) + 1 +--R (1) --------------------------------------- +--R a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.445~~~~~$\displaystyle +\int{\frac{dx}{\cot{ax}}}~dx$} +$$\int{\frac{1}{\cot{ax}}}= +-\frac{1}{a}\ln\cos{ax} +$$ +<<*>>= +)clear all + +--S 6 of 11 +aa:=integrate(1/cot(a*x),x) +--R +--R +--R 2 +--R log(-------------) +--R cos(2a x) + 1 +--R (1) ------------------ +--R 2a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.446~~~~~$\displaystyle +\int{x\cot{ax}}~dx$} +$$\int{x\cot{ax}}= +\frac{1}{a^2}\left\{ax +-\frac{(ax)^3}{9}-\frac{(ax)^5}{225} +-\cdots-\frac{2^{2n}B_n(ax)^{2n+1}}{(2n+1)!}-\cdots\right\} +$$ +<<*>>= +)clear all + +--S 7 of 11 +aa:=integrate(x*cot(a*x),x) +--R +--R +--R x +--R ++ +--I (1) | %I cot(%I a)d%I +--R ++ +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.447~~~~~$\displaystyle +\int{\frac{\cot{ax}}{x}}~dx$} +$$\int{\frac{\cot{ax}}{x}}= +-\frac{1}{ax}-\frac{ax}{3}-\frac{(ax)^3}{135}-\cdots +-\frac{2^{2n}B_n(ax)^{2n-1}}{(2n-1)(2n)!}-\cdots +$$ +<<*>>= +)clear all + +--S 8 of 11 +aa:=integrate(cot(a*x)/x,x) +--R +--R +--R x +--I ++ cot(%I a) +--I (1) | --------- d%I +--I ++ %I +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.448~~~~~$\displaystyle +\int{x\cot^2{ax}}~dx$} +$$\int{x\cot^2{ax}}= +-\frac{x\cot{ax}}{a}+\frac{1}{a^2}\ln\sin{ax}-\frac{x^2}{2} +$$ +<<*>>= +)clear all + +--S 9 of 11 +aa:=integrate(x*cot(a*x)^2,x) +--R +--R +--R (1) +--R sin(2a x) 2 +--R 2sin(2a x)log(-------------) - sin(2a x)log(-------------) +--R cos(2a x) + 1 cos(2a x) + 1 +--R + +--R 2 2 +--R - a x sin(2a x) - 2a x cos(2a x) - 2a x +--R / +--R 2 +--R 2a sin(2a x) +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.449~~~~~$\displaystyle +\int{\frac{dx}{p+q\cot{ax}}}~dx$} +$$\int{\frac{1}{p+q\cot{ax}}}= +\frac{px}{p^2+q^2}-\frac{q}{a(p^2+q^2)}\ln(p\sin{ax}+q\cos{ax}) +$$ +<<*>>= +)clear all + +--S 10 of 11 +aa:=integrate(1/(p+q*cot(a*x)),x) +--R +--R +--R (1) +--R p sin(2a x) + q cos(2a x) + q 2 +--R - 2q log(-----------------------------) + q log(-------------) + 2a p x +--R cos(2a x) + 1 cos(2a x) + 1 +--R ----------------------------------------------------------------------- +--R 2 2 +--R 2a q + 2a p +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.450~~~~~$\displaystyle +\int{\cot^n{ax}}~dx$} +$$\int{\cot^n{ax}}= +-\frac{\cot^{n-1}{ax}}{(n-1)a}-\int{\cos^{n-2}{ax}} +$$ +<<*>>= +)clear all + +--S 11 of 11 +aa:=integrate(cot(a*x)^n,x) +--R +--R +--R x +--R ++ n +--I (1) | cot(%I a) d%I +--R ++ +--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 p81 +\end{thebibliography} +\end{document} diff --git a/src/input/schaum22.input.pamphlet b/src/input/schaum22.input.pamphlet new file mode 100644 index 0000000..66befb9 --- /dev/null +++ b/src/input/schaum22.input.pamphlet @@ -0,0 +1,254 @@ +\documentclass{article} +\usepackage{axiom} +\begin{document} +\title{\$SPAD/input schaum22.input} +\author{Timothy Daly} +\maketitle +\eject +\tableofcontents +\eject +\section{\cite{1}:14.451~~~~~$\displaystyle +\int{\sec{ax}}~dx$} +$$\int{\sec{ax}}= +\frac{1}{a}\ln(\sec{ax}+\tan{ax})= +\frac{1}{a}\ln\tan\left(\frac{ax}{2}+\frac{\pi}{4}\right) +$$ +<<*>>= +)spool schaum22.output +)set message test on +)set message auto off +)clear all + +--S 1 of 10 +aa:=integrate(sec(a*x),x) +--R +--R +--R sin(a x) + cos(a x) + 1 sin(a x) - cos(a x) - 1 +--R log(-----------------------) - log(-----------------------) +--R cos(a x) + 1 cos(a x) + 1 +--R (1) ----------------------------------------------------------- +--R a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.452~~~~~$\displaystyle +\int{\sec^2{ax}}~dx$} +$$\int{\sec^2{ax}}= +\frac{\tan{ax}}{a} +$$ +<<*>>= +)clear all + +--S 2 of 10 +aa:=integrate(sec(a*x)^2,x) +--R +--R +--R sin(a x) +--R (1) ---------- +--R a cos(a x) +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.453~~~~~$\displaystyle +\int{\sec^3{ax}}~dx$} +$$\int{\sec^3{ax}}= +\frac{\sec{ax}\tan{ax}}{2a}+\frac{1}{2a}\ln(\sec{ax}+\tan{ax}) +$$ +<<*>>= +)clear all + +--S 3 of 10 +aa:=integrate(sec(a*x)^3,x) +--R +--R +--R (1) +--R 2 sin(a x) + cos(a x) + 1 +--R cos(a x) log(-----------------------) +--R cos(a x) + 1 +--R + +--R 2 sin(a x) - cos(a x) - 1 +--R - cos(a x) log(-----------------------) + sin(a x) +--R cos(a x) + 1 +--R / +--R 2 +--R 2a cos(a x) +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.454~~~~~$\displaystyle +\int{\sec^n{ax}\tan{ax}}~dx$} +$$\int{\sec^n{ax}\tan{ax}}= +\frac{\sec^n{ax}}{na} +$$ +<<*>>= +)clear all + +--S 4 of 10 +aa:=integrate(sec(a*x)^n*tan(a*x),x) +--R +--R 1 +--R n log(---------) +--R 2 +--R cos(a x) +--R ---------------- +--R 2 +--R %e +--R (1) ------------------ +--R a n +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.455~~~~~$\displaystyle +\int{\frac{dx}{\sec{ax}}}~dx$} +$$\int{\frac{1}{\sec{ax}}}= +\frac{\sin{ax}}{a} +$$ +<<*>>= +)clear all + +--S 5 of 10 +aa:=integrate(1/sec(a*x),x) +--R +--R +--R sin(a x) +--R (1) -------- +--R a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.456~~~~~$\displaystyle +\int{x\sec{ax}}~dx$} +$$\int{x\sec{ax}}= +\frac{1}{a^2}\left\{\frac{(ax)^2}{2}+\frac{(ax)^4}{8}+\frac{5(ax)^6}{144} ++\cdots+\frac{E_n(ax)^{2n+2}}{(2n+2)(2n)!}+\cdots\right\} +$$ +<<*>>= +)clear all + +--S 6 of 10 +aa:=integrate(x*sec(a*x),x) +--R +--R +--R x +--R ++ +--I (1) | %N sec(%N a)d%N +--R ++ +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.457~~~~~$\displaystyle +\int{\frac{\sec{ax}}{x}}~dx$} +$$\int{\frac{\sec{ax}}{x}}= +\ln{x}+\frac{(ax)^2}{4}+\frac{5(ax)^4}{96}+\frac{61(ax)^6}{4320} ++\cdots+\frac{E_n(ax)^{2n}}{(2n)(2n)!}+\cdots +$$ +<<*>>= +)clear all + +--S 7 of 10 +aa:=integrate(sec(a*x)/x,x) +--R +--R +--R x +--I ++ sec(%N a) +--I (1) | --------- d%N +--I ++ %N +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.458~~~~~$\displaystyle +\int{x\sec^2{ax}}~dx$} +$$\int{x\sec^2{ax}}= +\frac{x}{a}\tan{ax}+\frac{1}{a^2}\ln\cos{ax} +$$ +<<*>>= +)clear all + +--S 8 of 10 +aa:=integrate(x*sec(a*x)^2,x) +--R +--R +--R (1) +--R 2 2cos(a x) +--R - cos(a x)log(------------) + cos(a x)log(- ------------) + a x sin(a x) +--R cos(a x) + 1 cos(a x) + 1 +--R ------------------------------------------------------------------------ +--R 2 +--R a cos(a x) +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.459~~~~~$\displaystyle +\int{\frac{dx}{q+p\sec{ax}}}~dx$} +$$\int{\frac{1}{q+p\sec{ax}}}= +\frac{x}{q}-\frac{p}{q}\int{\frac{dx}{p+q\cos{ax}}} +$$ +<<*>>= +)clear all + +--S 9 of 10 +aa:=integrate(1/(q+p*sec(a*x)),x) +--R +--R +--R (1) +--R +-------+ +--R | 2 2 2 2 +-------+ +--R (- p cos(a x) - q)\|q - p + (q - p )sin(a x) | 2 2 +--R p log(------------------------------------------------) + a x\|q - p +--R q cos(a x) + p +--R [-----------------------------------------------------------------------, +--R +-------+ +--R | 2 2 +--R a q\|q - p +--R +---------+ +--R | 2 2 +---------+ +--R sin(a x)\|- q + p | 2 2 +--R - 2p atan(-----------------------) + a x\|- q + p +--R (q + p)cos(a x) + q + p +--R ----------------------------------------------------] +--R +---------+ +--R | 2 2 +--R a q\|- q + p +--R Type: Union(List Expression Integer,...) +--E +@ + +\section{\cite{1}:14.460~~~~~$\displaystyle +\int{\sec^n{ax}}~dx$} +$$\int{\sec^n{ax}}= +\frac{\sec^{n-2}{ax}\tan{ax}}{a(n-1)} ++\frac{n-2}{n-1}\int{\sec^{n-2}{ax}} +$$ +<<*>>= +)clear all + +--S 10 of 10 +aa:=integrate(sec(a*x)^n,x) +--R +--R +--R x +--R ++ n +--I (1) | sec(%N a) d%N +--R ++ +--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 pp81-82 +\end{thebibliography} +\end{document} diff --git a/src/input/schaum23.input.pamphlet b/src/input/schaum23.input.pamphlet new file mode 100644 index 0000000..679e775 --- /dev/null +++ b/src/input/schaum23.input.pamphlet @@ -0,0 +1,262 @@ +\documentclass{article} +\usepackage{axiom} +\begin{document} +\title{\$SPAD/input schaum23.input} +\author{Timothy Daly} +\maketitle +\eject +\tableofcontents +\eject +\section{\cite{1}:14.461~~~~~$\displaystyle +\int{\csc{ax}}~dx$} +$$\int{\csc{ax}}= +\frac{1}{a}\ln(\csc{ax}-\cot{ax})= +\frac{1}{a}\ln\tan{\frac{ax}{2}} +$$ +<<*>>= +)spool schaum23.output +)set message test on +)set message auto off +)clear all + +--S 1 of 10 +aa:=integrate(csc(a*x),x) +--R +--R +--R sin(a x) +--R log(------------) +--R cos(a x) + 1 +--R (1) ----------------- +--R a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.462~~~~~$\displaystyle +\int{\csc^2{ax}}~dx$} +$$\int{\csc^2{ax}}= +-\frac{\cot{ax}}{a} +$$ +<<*>>= +)clear all + +--S 2 of 10 +aa:=integrate(csc(a*x)^2,x) +--R +--R +--R cos(a x) +--R (1) - ---------- +--R a sin(a x) +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.463~~~~~$\displaystyle +\int{\csc^3{ax}}~dx$} +$$\int{\csc^3{ax}}= +-\frac{\csc{ax}\cot{ax}}{2a}+\frac{1}{2a}\ln\tan{\frac{ax}{2}} +$$ +<<*>>= +)clear all + +--S 3 of 10 +aa:=integrate(csc(a*x)^3,x) +--R +--R +--R 2 sin(a x) +--R (cos(a x) - 1)log(------------) + cos(a x) +--R cos(a x) + 1 +--R (1) ------------------------------------------- +--R 2 +--R 2a cos(a x) - 2a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.464~~~~~$\displaystyle +\int{\csc^n{ax}\cot{ax}}~dx$} +$$\int{\csc^n{ax}\cot{ax}}= +-\frac{csc^n{ax}}{na} +$$ +<<*>>= +)clear all + +--S 4 of 10 +aa:=integrate(csc(a*x)^n*cot(a*x),x) +--R +--R +--R 1 +--R n log(- -------------) +--R 2 +--R cos(a x) - 1 +--R ---------------------- +--R 2 +--R %e +--R (1) - ------------------------ +--R a n +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.465~~~~~$\displaystyle +\int{\frac{dx}{\csc{ax}}}~dx$} +$$\int{\frac{1}{\csc{ax}}}= +-\frac{\cos{ax}}{a} +$$ +<<*>>= +)clear all + +--S 5 of 10 +aa:=integrate(1/csc(a*x),x) +--R +--R +--R cos(a x) +--R (1) - -------- +--R a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.466~~~~~$\displaystyle +\int{x\csc{ax}}~dx$} +$$\int{x\csc{ax}}= +\frac{1}{a^2}\left\{ax+\frac{(ax)^3}{18}+\frac{7(ax)^5}{1800} ++\cdots+\frac{2(2^{2n-1}-1)B_n(ax)^{2n+1}}{(2n+1)!}+\cdots\right\} +$$ +<<*>>= +)clear all + +--S 6 of 10 +aa:=integrate(x*csc(a*x),x) +--R +--R +--R x +--R ++ +--I (1) | %H csc(%H a)d%H +--R ++ +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.467~~~~~$\displaystyle +\int{\frac{\csc{ax}}{x}}~dx$} +$$\int{\frac{\csc{ax}}{x}}= +-\frac{1}{ax}+\frac{(ax)}{6}+\frac{7(ax)^3}{1800} ++\cdots+\frac{2(2^{2n-1}-1)B_n(ax)^{2n-1}}{(2n-1)(2n)!}+\cdots +$$ +<<*>>= +)clear all + +--S 7 of 10 +aa:=integrate(csc(a*x)/x,x) +--R +--R +--R x +--I ++ csc(%H a) +--I (1) | --------- d%H +--I ++ %H +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.468~~~~~$\displaystyle +\int{x\csc^2{ax}}~dx$} +$$\int{x\csc^2{ax}}= +-\frac{x\cot{ax}}{a}+\frac{1}{a^2}\ln\sin{ax} +$$ +<<*>>= +)clear all + +--S 8 of 10 +aa:=integrate(x*csc(a*x)^2,x) +--R +--R +--R sin(a x) 2 +--R sin(a x)log(------------) - sin(a x)log(------------) - a x cos(a x) +--R cos(a x) + 1 cos(a x) + 1 +--R (1) -------------------------------------------------------------------- +--R 2 +--R a sin(a x) +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.469~~~~~$\displaystyle +\int{\frac{dx}{q+p\csc{ax}}}~dx$} +$$\int{\frac{1}{q+p\csc{ax}}}= +\frac{x}{q}-\frac{p}{q}\int{\frac{1}{p+q\sin{ax}}} +$$ +<<*>>= +)clear all + +--S 9 of 10 +aa:=integrate(1/(q+p*csc(a*x)),x) +--R +--R +--R (1) +--R [ +--R p +--R * +--R log +--R +-------+ +--R 2 2 2 | 2 2 +--R (p q sin(a x) + (q - p )cos(a x) + q )\|q - p +--R + +--R 2 3 3 2 3 2 +--R (p q - p )sin(a x) + (q - p q)cos(a x) + q - p q +--R / +--R q sin(a x) + p +--R + +--R +-------+ +--R | 2 2 +--R a x\|q - p +--R / +--R +-------+ +--R | 2 2 +--R a q\|q - p +--R , +--R +---------+ +--R | 2 2 +---------+ +--R (p sin(a x) + q cos(a x) + q)\|- q + p | 2 2 +--R 2p atan(-----------------------------------------) + a x\|- q + p +--R 2 2 2 2 +--R (q - p )cos(a x) + q - p +--R --------------------------------------------------------------------] +--R +---------+ +--R | 2 2 +--R a q\|- q + p +--R Type: Union(List Expression Integer,...) +--E +@ + +\section{\cite{1}:14.470~~~~~$\displaystyle +\int{\csc^n{ax}}~dx$} +$$\int{\csc^n{ax}}= +-\frac{\csc^{n-2}{ax}\cot{ax}}{a(n-1)} ++\frac{n-2}{n-1}\int{\csc^{n-2}{ax}} +$$ +<<*>>= +)clear all + +--S 10 of 10 +aa:=integrate(csc(a*x)^n,x) +--R +--R +--R x +--R ++ n +--I (1) | csc(%H a) d%H +--R ++ +--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 p82 +\end{thebibliography} +\end{document} diff --git a/src/input/schaum24.input.pamphlet b/src/input/schaum24.input.pamphlet new file mode 100644 index 0000000..1a6b60a --- /dev/null +++ b/src/input/schaum24.input.pamphlet @@ -0,0 +1,1036 @@ +\documentclass{article} +\usepackage{axiom} +\begin{document} +\title{\$SPAD/input schaum24.input} +\author{Timothy Daly} +\maketitle +\eject +\tableofcontents +\eject +\section{\cite{1}:14.471~~~~~$\displaystyle +\int{\sin^{-1}{\frac{x}{a}}}~dx$} +$$\int{\sin^{-1}{\frac{x}{a}}}= +x\sin^{-1}{\frac{x}{a}}+\sqrt{a^2-x^2} +$$ +<<*>>= +)spool schaum24.output +)set message test on +)set message auto off +)clear all + +--S 1 of 38 +aa:=integrate(asin(x/a),x) +--R +--R +--R +---------+ +--R | 2 2 +---------+ +--R 2x\|- x + a | 2 2 +--R - x atan(--------------) + 2\|- x + a +--R 2 2 +--R 2x - a +--R (1) ---------------------------------------- +--R 2 +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.472~~~~~$\displaystyle +\int{x\sin^{-1}{\frac{x}{a}}}~dx$} +$$\int{x\sin^{-1}{\frac{x}{a}}}= +\left(\frac{x^2}{2}-\frac{a^2}{4}\right)\sin^{-1}{\frac{x}{a}} ++\frac{x\sqrt{a^2-x^2}}{4} +$$ +<<*>>= +)clear all + +--S 2 of 38 +aa:=integrate(x*asin(x/a),x) +--R +--R +--R +---------+ +--R | 2 2 +---------+ +--R 2 2 2x\|- x + a | 2 2 +--R (- 2x + a )atan(--------------) + 2x\|- x + a +--R 2 2 +--R 2x - a +--R (1) ------------------------------------------------- +--R 8 +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.473~~~~~$\displaystyle +\int{x^2\sin^{-1}\frac{x}{a}}~dx$} +$$\int{x^2\sin^{-1}\frac{x}{a}}= +\frac{x^3}{3}\sin^{-1}\frac{x}{a}+\frac{(x^2+2a^2)\sqrt{a^2-x^2}}{9} +$$ +<<*>>= +)clear all + +--S 3 of 38 +aa:=integrate(x^2*asin(x/a),x) +--R +--R +--R +---------+ +--R | 2 2 +---------+ +--R 3 2x\|- x + a 2 2 | 2 2 +--R - 3x atan(--------------) + (2x + 4a )\|- x + a +--R 2 2 +--R 2x - a +--R (1) --------------------------------------------------- +--R 18 +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.474~~~~~$\displaystyle +\int{\frac{\sin^{-1}(x/a)}{x}}~dx$} +$$\int{\frac{\sin^{-1}(x/a)}{x}}= +\frac{x}{a}+\frac{(x/a)^3}{2\cdot 3\cdot 3} ++\frac{1\cdot 3(x/a)^5}{2\cdot 4\cdot 5\cdot 5} ++\frac{1\cdot 3\cdot 5(x/a)^7}{2\cdot 4\cdot 6\cdot 7\cdot 7}+\cdots +$$ +<<*>>= +)clear all + +--S 4 of 38 +aa:=integrate(asin(x/a)/x,x) +--R +--R +--I %H +--R x asin(--) +--R ++ a +--I (1) | -------- d%H +--I ++ %H +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.475~~~~~$\displaystyle +\int{\frac{\sin^{-1}{(x/a)}}{x^2}}~dx$} +$$\int{\frac{\sin^{-1}{(x/a)}}{x^2}}= +-\frac{\sin^{-1}(x/a)}{x} +-\frac{1}{a}\ln\left(\frac{a+\sqrt{a^2-x^2}}{x}\right) +$$ +<<*>>= +)clear all + +--S 5 of 38 +aa:=integrate(asin(x/a)/x^2,x) +--R +--R +--R (1) +--R +---------+ +--R +---------+ +---------+ | 2 2 +--R | 2 2 | 2 2 2x\|- x + a +--R - x log(\|- x + a + a) + x log(\|- x + a - a) + a atan(--------------) +--R 2 2 +--R 2x - a +--R ---------------------------------------------------------------------------- +--R 2a x +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.476~~~~~$\displaystyle +\int{\left(sin^{-1}\frac{x}{a}\right)^2}~dx$} +$$\int{\left(sin^{-1}\frac{x}{a}\right)^2}= +x\left(\sin^{-1}\frac{x}{a}\right)^2-2x+2\sqrt{a^2-x^2}\sin^{-1}\frac{x}{a} +$$ +<<*>>= +)clear all + +--S 6 of 38 +aa:=integrate(asin(x/a)^2,x) +--R +--R +--R +---------+ 2 +---------+ +--R | 2 2 +---------+ | 2 2 +--R 2x\|- x + a | 2 2 2x\|- x + a +--R x atan(--------------) - 4\|- x + a atan(--------------) - 8x +--R 2 2 2 2 +--R 2x - a 2x - a +--R (1) ---------------------------------------------------------------- +--R 4 +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.477~~~~~$\displaystyle +\int{\cos^{-1}\frac{x}{a}}~dx$} +$$\int{\cos^{-1}\frac{x}{a}}= +x\cos^{-1}\frac{x}{a}-\sqrt{a^2-x^2} +$$ +<<*>>= +)clear all + +--S 7 of 38 +aa:=integrate(acos(x/a),x) +--R +--R +--R +---------+ +--R | 2 2 +---------+ +--R 2x\|- x + a | 2 2 +--R x atan(--------------) - 2\|- x + a +--R 2 2 +--R 2x - a +--R (1) -------------------------------------- +--R 2 +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.478~~~~~$\displaystyle +\int{x\cos^{-1}\frac{x}{a}}~dx$} +$$\int{x\cos^{-1}\frac{x}{a}}= +\left(\frac{x^2}{2}-\frac{a^2}{4}\right)\cos^{-1}\frac{x}{a} +-\frac{x\sqrt{a^2-x^2}}{4} +$$ +<<*>>= +)clear all + +--S 8 of 38 +aa:=integrate(x*acos(x/a),x) +--R +--R +--R +---------+ +--R | 2 2 +---------+ +--R 2 2 2x\|- x + a | 2 2 +--R (2x - a )atan(--------------) - 2x\|- x + a +--R 2 2 +--R 2x - a +--R (1) ----------------------------------------------- +--R 8 +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.479~~~~~$\displaystyle +\int{x^2\cos^{-1}\frac{x}{a}}~dx$} +$$\int{x^2\cos^{-1}\frac{x}{a}}= +\frac{x^3}{3}\cos^{-1}\frac{x}{a}-\frac{(x^2+2a^2)\sqrt{a^2-x^2}}{9} +$$ +<<*>>= +)clear all + +--S 9 of 38 +aa:=integrate(x^2*acos(x/a),x) +--R +--R +--R +---------+ +--R | 2 2 +---------+ +--R 3 2x\|- x + a 2 2 | 2 2 +--R 3x atan(--------------) + (- 2x - 4a )\|- x + a +--R 2 2 +--R 2x - a +--R (1) --------------------------------------------------- +--R 18 +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.480~~~~~$\displaystyle +\int{\frac{\cos^{-1}(x/a)}{x}}~dx$} +$$\int{\frac{\cos^{-1}(x/a)}{x}}= +\frac{x}{2}\ln{x}-\int{\frac{\sin^{-1}(x/a)}{x}} +$$ +<<*>>= +)clear all + +--S 10 of 38 +aa:=integrate(acos(x/a)/x,x) +--R +--R +--I %H +--R x acos(--) +--R ++ a +--I (1) | -------- d%H +--I ++ %H +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.481~~~~~$\displaystyle +\int{\frac{\cos^{-1}(x/a)}{x^2}}~dx$} +$$\int{\frac{\cos^{-1}(x/a)}{x^2}}= +-\frac{\cos^{-1}(x/a)}{x}+\frac{1}{a}\ln\left(\frac{a+\sqrt{a^2-x^2}}{x}\right) +$$ +<<*>>= +)clear all + +--S 11 of 38 +aa:=integrate(acos(x/a)/x^2,x) +--R +--R +--R (1) +--R +---------+ +--R +---------+ +---------+ | 2 2 +--R | 2 2 | 2 2 2x\|- x + a +--R x log(\|- x + a + a) - x log(\|- x + a - a) - a atan(--------------) +--R 2 2 +--R 2x - a +--R -------------------------------------------------------------------------- +--R 2a x +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.482~~~~~$\displaystyle +\int{\left(\cos^{-1}\frac{x}{a}\right)^2}~dx$} +$$\int{\left(\cos^{-1}\frac{x}{a}\right)^2}= +x\left(\cos^{-1}\frac{x}{a}\right)^2-2x-2\sqrt{a^2-x^2}\cos^{-1}\frac{x}{a} +$$ +<<*>>= +)clear all + +--S 12 of 38 +aa:=integrate(acos(x/a)^2,x) +--R +--R +--R +---------+ 2 +---------+ +--R | 2 2 +---------+ | 2 2 +--R 2x\|- x + a | 2 2 2x\|- x + a +--R x atan(--------------) - 4\|- x + a atan(--------------) - 8x +--R 2 2 2 2 +--R 2x - a 2x - a +--R (1) ---------------------------------------------------------------- +--R 4 +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.483~~~~~$\displaystyle +\int{\tan^{-1}\frac{x}{a}}~dx$} +$$\int{\tan^{-1}\frac{x}{a}}= +x\tan^{-1}\frac{x}{a}-\frac{a}{2}\ln(x^2+a^2) +$$ +<<*>>= +)clear all + +--S 13 of 38 +aa:=integrate(atan(x/a),x) +--R +--R +--R 2 2 2a x +--R - a log(x + a ) - x atan(-------) +--R 2 2 +--R x - a +--R (1) ---------------------------------- +--R 2 +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.484~~~~~$\displaystyle +\int{x\tan^{-1}\frac{x}{a}}~dx$} +$$\int{x\tan^{-1}\frac{x}{a}}= +\frac{1}{2}(x^2+a^2)\tan^{-1}\frac{x}{a}-\frac{ax}{2} +$$ +<<*>>= +)clear all + +--S 14 of 38 +aa:=integrate(x*tan(x/a),x) +--R +--R +--R x +--I ++ %H +--I (1) | %H tan(--)d%H +--R ++ a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.485~~~~~$\displaystyle +\int{x^2\tan^{-1}\frac{x}{a}}~dx$} +$$\int{x^2\tan^{-1}\frac{x}{a}}= +\frac{x^3}{3}\tan^{-1}\frac{x}{a}-\frac{ax^2}{6}+\frac{a^3}{6}\ln(x^2+a^2) +$$ +<<*>>= +)clear all + +--S 15 of 38 +aa:=integrate(x^2*atan(x/a),x) +--R +--R +--R 3 2 2 3 2a x 2 +--R a log(x + a ) - x atan(-------) - a x +--R 2 2 +--R x - a +--R (1) --------------------------------------- +--R 6 +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.486~~~~~$\displaystyle +\int{\frac{\tan^{-1}(x/a)}{x}}~dx$} +$$\int{\frac{\tan^{-1}(x/a)}{x}}= +\frac{x}{a}-\frac{(x/a)^3}{3^2}+\frac{(x/a)^5}{5^2}-\frac{(x/a)^7}{7^2}+\cdots +$$ +<<*>>= +)clear all + +--S 16 of 38 +aa:=integrate(atan(x/a)/x,x) +--R +--R +--I %H +--R x atan(--) +--R ++ a +--I (1) | -------- d%H +--I ++ %H +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.487~~~~~$\displaystyle +\int{\frac{\tan^{-1}(x/a)}{x^2}}~dx$} +$$\int{\frac{\tan^{-1}(x/a)}{x^2}}= +-\frac{1}{x}\tan^{-1}\frac{x}{a} +-\frac{1}{2a}\ln\left(\frac{x^2+a^2}{x^2}\right) +$$ +<<*>>= +)clear all + +--S 17 of 38 +aa:=integrate(atan(x/a)/x^2,x) +--R +--R +--R 2 2 2a x +--R - x log(x + a ) + 2x log(x) + a atan(-------) +--R 2 2 +--R x - a +--R (1) ---------------------------------------------- +--R 2a x +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.488~~~~~$\displaystyle +\int{\cot^{-1}\frac{x}{a}}~dx$} +$$\int{\cot^{-1}\frac{x}{a}}= +x\cot^{-1}\frac{x}{a}+\frac{a}{2}\ln(x^2+a^2) +$$ +<<*>>= +)clear all + +--S 18 of 38 +aa:=integrate(acot(x/a),x) +--R +--R +--R 2 2 2a x +--R a log(x + a ) + x atan(-------) +--R 2 2 +--R x - a +--R (1) -------------------------------- +--R 2 +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.489~~~~~$\displaystyle +\int{x\cot^{-1}\frac{x}{a}}~dx$} +$$\int{x\cot^{-1}\frac{x}{a}}= +\frac{1}{2}(x^2+a^2)\cot^{-1}\frac{x}{a}+\frac{ax}{2} +$$ +<<*>>= +)clear all + +--S 19 of 38 +aa:=integrate(x*acot(x/a),x) +--R +--R +--R 2 2 2a x +--R (x + a )atan(-------) + 2a x +--R 2 2 +--R x - a +--R (1) ----------------------------- +--R 4 +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.490~~~~~$\displaystyle +\int{x^2\cot^{-1}\frac{x}{a}}~dx$} +$$\int{x^2\cot^{-1}\frac{x}{a}}= +\frac{x^3}{3}\cot^{-1}\frac{x}{a}+\frac{ax^2}{6}-\frac{a^3}{6}\ln(x^2+a^2) +$$ +<<*>>= +)clear all + +--S 20 of 38 +aa:=integrate(x^2*acot(x/a),x) +--R +--R +--R 3 2 2 3 2a x 2 +--R - a log(x + a ) + x atan(-------) + a x +--R 2 2 +--R x - a +--R (1) ----------------------------------------- +--R 6 +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.491~~~~~$\displaystyle +\int{\frac{\cot^{-1}(x/a)}{x}}~dx$} +$$\int{\frac{\cot^{-1}(x/a)}{x}}= +\frac{\pi}{2}\ln{x}-\int{\frac{\tan^{-1}(x/a)}{x}} +$$ +<<*>>= +)clear all + +--S 21 of 38 +aa:=integrate(acot(x/a)/x,x) +--R +--R +--I %H +--R x acot(--) +--R ++ a +--I (1) | -------- d%H +--I ++ %H +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.492~~~~~$\displaystyle +\int{\frac{\cot^{-1}(x/a)}{x^2}}~dx$} +$$\int{\frac{\cot^{-1}(x/a)}{x^2}}= +-\frac{cot^{-1}(x/a)}{x}+\frac{1}{2a}\ln\left(\frac{x^2+a^2}{x^2}\right) +$$ +<<*>>= +)clear all + +--S 22 of 38 +aa:=integrate(acot(x/a)/x^2,x) +--R +--R +--R 2 2 2a x +--R x log(x + a ) - 2x log(x) - a atan(-------) +--R 2 2 +--R x - a +--R (1) -------------------------------------------- +--R 2a x +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.493~~~~~$\displaystyle +\int{\sec^{-1}\frac{x}{a}}~dx$} +$$\int{\sec^{-1}\frac{x}{a}}= +\left\{ +\begin{array}{l} +\displaystyle +x\sec^{-1}\frac{x}{a}-a\ln(x+\sqrt{x^2-a^2}) +{\rm \ if\ }0 < \sec^{-1}\frac{x}{a} < \frac{\pi}{2}\\ +\\ +\displaystyle +x\sec^{-1}\frac{x}{a}+a\ln(x+\sqrt{x^2-a^2}) +{\rm \ if\ }\frac{\pi}{2} < \sec^{-1}\frac{x}{a} < \pi +\end{array} +\right. +$$ +<<*>>= +)clear all + +--S 23 of 38 +aa:=integrate(asec(x/a),x) +--R +--R +--R (1) +--R +---------+ +---------+ +--R +-+ | 2 2 | 2 2 +--R +-+ 2x\|2 \|- x + a 2a\|- x + a +--R - a\|2 atan(------------------) + x atan(--------------) +--R 2 2 2 +--R 3x - 2a x +--R + +--R x +--R - 2a atan(------------) +--R +---------+ +--R | 2 2 +--R \|- x + a +--R / +--R 2 +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.494~~~~~$\displaystyle +\int{x\sec^{-1}\frac{x}{a}}~dx$} +$$\int{x\sec^{-1}\frac{x}{a}}= +\left\{ +\begin{array}{l} +\displaystyle +\frac{x^2}{2}\sec^{-1}\frac{x}{a}-\frac{a\sqrt{x^2-a^2}}{2} +{\rm \ if\ }0 < \sec^{-1}\frac{x}{a} < \frac{\pi}{2}\\ +\\ +\displaystyle +\frac{x^2}{2}\sec^{-1}\frac{x}{a}+\frac{a\sqrt{x^2-a^2}}{2} +{\rm \ if\ }\frac{\pi}{2} < \sec^{-1}\frac{x}{a} < \pi +\end{array} +\right. +$$ +<<*>>= +)clear all + +--S 24 of 38 +aa:=integrate(x*asec(x/a),x) +--R +--R +--R +---------+ +--R | 2 2 +---------+ +--R 2 2 2a\|- x + a | 2 2 +--R (x - 2a )atan(--------------) + 2a\|- x + a +--R 2 +--R x +--R (1) ----------------------------------------------- +--R 4 +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.495~~~~~$\displaystyle +\int{x^2\sec^{-1}\frac{x}{a}}~dx$} +$$\int{x^2\sec^{-1}\frac{x}{a}}= +\left\{ +\begin{array}{l} +\displaystyle +\frac{x^3}{3}\sec^{-1}\frac{x}{a}-\frac{ax\sqrt{x^2-a^2}}{6} +-\frac{a^3}{6}\ln(x+\sqrt{x^2-a^2})\\ +\\ +\displaystyle +\hbox{\hskip 3cm}{\rm \ if\ }0 < \sec^{-1}\frac{x}{a} < \frac{\pi}{2}\\ +\\ +\displaystyle +\frac{x^3}{3}\sec^{-1}\frac{x}{a}+\frac{ax\sqrt{x^2-a^2}}{6} ++\frac{a^3}{6}\ln(x+\sqrt{x^2-a^2})\\ +\\ +\displaystyle +\hbox{\hskip 3cm} +{\rm \ if\ }\frac{\pi}{2} < \sec^{-1}\frac{x}{a} < \pi\\ +\end{array} +\right. +$$ +<<*>>= +)clear all + +--S 25 of 38 +aa:=integrate(x^2*asec(x/a),x) +--R +--R +--R (1) +--R +---------+ +---------+ +--R +-+ | 2 2 | 2 2 +--R 3 +-+ 2x\|2 \|- x + a 3 2a\|- x + a +--R - 2a \|2 atan(------------------) + x atan(--------------) +--R 2 2 2 +--R 3x - 2a x +--R + +--R +---------+ +--R 3 x | 2 2 +--R - 5a atan(------------) + a x\|- x + a +--R +---------+ +--R | 2 2 +--R \|- x + a +--R / +--R 6 +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.496~~~~~$\displaystyle +\int{\frac{\sec^{-1}(x/a)}{x}}~dx$} +$$\int{\frac{\sec^{-1}(x/a)}{x}}= +\frac{\pi}{2}\ln{x}+\frac{a}{x}+\frac{(a/x)^3}{2\cdot 3\cdot 3} ++\frac{1\cdot 3(a/x)^5}{2\cdot 4\cdot 5\cdot 5} ++\frac{1\cdot 3\cdot 5(a/x)^7}{2\cdot 4\cdot 6\cdot 7\cdot 7}+\cdots +$$ +<<*>>= +)clear all + +--S 26 of 38 +aa:=integrate(asec(x/a)/x,x) +--R +--R +--I %H +--R x asec(--) +--R ++ a +--I (1) | -------- d%H +--I ++ %H +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.497~~~~~$\displaystyle +\int{\frac{\sec^{-1}(x/a)}{x^2}}~dx$} +$$\int{\frac{\sec^{-1}(x/a)}{x^2}}= +\left\{ +\begin{array}{l} +\displaystyle +-\frac{\sec^{-1}(x/a)}{x}+\frac{\sqrt{x^2-a^2}}{ax} +{\rm \ if\ }0 < \sec^{-1}\frac{x}{a} < \frac{\pi}{2}\\ +\\ +\displaystyle +-\frac{\sec^{-1}(x/a)}{x}-\frac{\sqrt{x^2-a^2}}{ax} +{\rm \ if\ }\frac{\pi}{2} < \sec^{-1}\frac{x}{a} < \pi\\ +\end{array} +\right. +$$ +<<*>>= +)clear all + +--S 27 of 38 +aa:=integrate(asec(x/a)/x^2,x) +--R +--R +--R +---------+ +---------+ +--R +-+ | 2 2 | 2 2 +--R 2x\|2 \|- x + a +-+ 2a\|- x + a +--R x atan(------------------) - a\|2 atan(--------------) +--R 2 2 2 +--R 3x - 2a x +--R (1) ------------------------------------------------------ +--R +-+ +--R 2a x\|2 +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.498~~~~~$\displaystyle +\int{\csc^{-1}\frac{x}{a}}~dx$} +$$\int{\csc^{-1}\frac{x}{a}}= +\left\{ +\begin{array}{l} +\displaystyle +x\csc^{-1}\frac{x}{a}+a\ln(x+\sqrt{x^2-a^2}) +{\rm \ if\ }0 < \csc^{-1}\frac{x}{a} < \frac{\pi}{2}\\ +\\ +\displaystyle +x\csc^{-1}\frac{x}{a}-a\ln(x+\sqrt{x^2-a^2}) +{\rm \ if\ }-\frac{\pi}{2} < \csc^{-1}\frac{x}{a} < 0\\ +\end{array} +\right. +$$ +<<*>>= +)clear all + +--S 28 of 38 +aa:=integrate(acsc(x/a),x) +--R +--R +--R (1) +--R +---------+ +---------+ +--R +-+ | 2 2 | 2 2 +--R +-+ 2x\|2 \|- x + a 2a\|- x + a +--R a\|2 atan(------------------) - x atan(--------------) +--R 2 2 2 +--R 3x - 2a x +--R + +--R x +--R 2a atan(------------) +--R +---------+ +--R | 2 2 +--R \|- x + a +--R / +--R 2 +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.499~~~~~$\displaystyle +\int{x\csc^{-1}\frac{x}{a}}~dx$} +$$\int{x\csc^{-1}\frac{x}{a}}= +\left\{ +\begin{array}{l} +\displaystyle +\frac{x^2}{2}\csc^{-1}\frac{x}{a}+\frac{a\sqrt{x^2-a^2}}{2} +{\rm \ if\ }0 < \csc^{-1}\frac{x}{a} < \frac{\pi}{2}\\ +\\ +\displaystyle +\frac{x^2}{2}\csc^{-1}\frac{x}{a}-\frac{a\sqrt{x^2-a^2}}{2} +{\rm \ if\ }-\frac{\pi}{2} < \csc^{-1}\frac{x}{a} < 0\\ +\end{array} +\right. +$$ +<<*>>= +)clear all + +--S 29 of 38 +aa:=integrate(x*acsc(x/a),x) +--R +--R +--R +---------+ +--R | 2 2 +---------+ +--R 2 2 2a\|- x + a | 2 2 +--R (- x + 2a )atan(--------------) - 2a\|- x + a +--R 2 +--R x +--R (1) ------------------------------------------------- +--R 4 +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.500~~~~~$\displaystyle +\int{x^2\csc^{-1}\frac{x}{a}}~dx$} +$$\int{x^2\csc^{-1}\frac{x}{a}}= +\left\{ +\begin{array}{l} +\displaystyle +\frac{x^3}{3}\csc^{-1}\frac{x}{a}+\frac{ax\sqrt{x^2-a^2}}{6} ++\frac{a^3}{6}\ln(x+\sqrt{x^2-a^2})\\ +\\ +\displaystyle +\hbox{\hskip 3cm}{\rm \ if\ }0 < \csc^{-1}\frac{x}{a} < \frac{\pi}{2}\\ +\\ +\displaystyle +\frac{x^3}{3}\sec^{-1}\frac{x}{a}-\frac{ax\sqrt{x^2-a^2}}{6} +-\frac{a^3}{6}\ln(x+\sqrt{x^2-a^2})\\ +\\ +\displaystyle +\hbox{\hskip 3cm} +{\rm \ if\ }-\frac{\pi}{2} < \csc^{-1}\frac{x}{a} < 0\\ +\end{array} +\right. +$$ +<<*>>= +)clear all + +--S 30 of 38 +aa:=integrate(x^2*acsc(x/a),x) +--R +--R +--R (1) +--R +---------+ +---------+ +--R +-+ | 2 2 | 2 2 +--R 3 +-+ 2x\|2 \|- x + a 3 2a\|- x + a +--R 2a \|2 atan(------------------) - x atan(--------------) +--R 2 2 2 +--R 3x - 2a x +--R + +--R +---------+ +--R 3 x | 2 2 +--R 5a atan(------------) - a x\|- x + a +--R +---------+ +--R | 2 2 +--R \|- x + a +--R / +--R 6 +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.501~~~~~$\displaystyle +\int{\frac{\csc^{-1}(x/a)}{x}}~dx$} +$$\int{\frac{\csc^{-1}(x/a)}{x}}= +-\left(\frac{a}{x}+\frac{(a/x)^3}{2\cdot 3\cdot 3} ++\frac{1\cdot 3(a/x)^5}{2\cdot 4\cdot 5\cdot 5} ++\frac{1\cdot 3\cdot 5(a/x)^7}{2\cdot 4\cdot 6\cdot 7\cdot 7}+\cdots\right) +$$ +<<*>>= +)clear all + +--S 31 of 38 +aa:=integrate(acsc(x/a)/x,x) +--R +--R +--I %H +--R x acsc(--) +--R ++ a +--I (1) | -------- d%H +--I ++ %H +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.502~~~~~$\displaystyle +\int{\frac{\csc^{-1}(x/a)}{x^2}}~dx$} +$$\int{\frac{\csc^{-1}(x/a)}{x^2}}= +\left\{ +\begin{array}{l} +\displaystyle +-\frac{csc^{-1}(x/a)}{x}-\frac{\sqrt{x^2-a^2}}{ax} +{\rm \ if\ }0 < \csc^{-1}\frac{x}{a} < \frac{\pi}{2}\\ +\\ +\displaystyle +-\frac{csc^{-1}(x/a)}{x}+\frac{\sqrt{x^2-a^2}}{ax} +{\rm \ if\ }-\frac{\pi}{2} < \csc^{-1}\frac{x}{a} < 0\\ +\end{array} +\right. +$$ +<<*>>= +)clear all + +--S 32 of 38 +aa:=integrate(acsc(x/a)/x^2,x) +--R +--R +--R +---------+ +---------+ +--R +-+ | 2 2 | 2 2 +--R 2x\|2 \|- x + a +-+ 2a\|- x + a +--R - x atan(------------------) + a\|2 atan(--------------) +--R 2 2 2 +--R 3x - 2a x +--R (1) -------------------------------------------------------- +--R +-+ +--R 2a x\|2 +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.503~~~~~$\displaystyle +\int{x^m\sin^{-1}\frac{x}{a}}~dx$} +$$\int{x^m\sin^{-1}\frac{x}{a}}= +\frac{x^{m+1}}{m+1}\sin^{-1}\frac{x}{a}-\frac{1}{m+1}\int{\frac{x^{m+1}}{\sqrt{a^2-x^2}}} +$$ +<<*>>= +)clear all + +--S 33 of 38 +aa:=integrate(x^m*asin(x/a),x) +--R +--R +--R x +--I ++ %H m +--I (1) | asin(--)%H d%H +--R ++ a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.504~~~~~$\displaystyle +\int{x^m\cos^{-1}\frac{x}{a}}~dx$} +$$\int{x^m\cos^{-1}\frac{x}{a}}= +\frac{x^{m+1}}{m+1}\cos^{-1}\frac{x}{a}+\frac{1}{m+1}\int{\frac{x^{m+1}}{\sqrt{a^2-x^2}}} +$$ +<<*>>= +)clear all + +--S 34 of 38 +aa:=integrate(x^m*acos(x/a),x) +--R +--R +--R x +--I ++ %H m +--I (1) | acos(--)%H d%H +--R ++ a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.505~~~~~$\displaystyle +\int{x^m\tan^{-1}\frac{x}{a}}~dx$} +$$\int{x^m\tan^{-1}\frac{x}{a}}= +\frac{x^{m_1}}{m+1}\tan^{-1}\frac{x}{a} +-\frac{a}{m+1}\int{\frac{x^{m+1}}{x^2+a^2}} +$$ +<<*>>= +)clear all + +--S 35 of 38 +aa:=integrate(x*m*atan(x/a),x) +--R +--R +--R 2 2 2a x +--R (- m x - a m)atan(-------) - 2a m x +--R 2 2 +--R x - a +--R (1) ------------------------------------ +--R 4 +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.506~~~~~$\displaystyle +\int{x^m\cot^{-1}\frac{x}{a}}~dx$} +$$\int{x^m\cot^{-1}\frac{x}{a}}= +\frac{x^{m+1}}{m+1}\cot^{-1}\frac{x}{a} ++\frac{a}{m+1}\int{\frac{x^{m+1}}{x^2+a^2}} +$$ +<<*>>= +)clear all + +--S 36 of 38 +aa:=integrate(x^m*acot(x/a),x) +--R +--R +--R x +--I ++ %H m +--I (1) | acot(--)%H d%H +--R ++ a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.507~~~~~$\displaystyle +\int{x^m\sec^{-1}\frac{x}{a}}~dx$} +$$\int{x^m\sec^{-1}\frac{x}{a}}= +\left\{ +\begin{array}{l} +\displaystyle +\frac{x^{m+1}\sec^{-1}(x/a)}{m+1}-\frac{a}{m+1}\int{\frac{x^m}{\sqrt{x^2-a^2}}} +{\rm \ if\ }0 < \sec^{-1}\frac{x}{a} < \frac{\pi}{2}\\ +\\ +\displaystyle +\frac{x^{m+1}\sec^{-1}(x/a)}{m+1}+\frac{a}{m+1}\int{\frac{x^m}{\sqrt{x^2-a^2}}} +{\rm \ if\ }\frac{\pi}{2} < \sec^{-1}\frac{x}{a} < \pi\\ +\end{array} +\right. +$$ +<<*>>= +)clear all + +--S 37 of 38 +aa:=integrate(x^m*asec(x/a),x) +--R +--R +--R x +--I ++ %H m +--I (1) | asec(--)%H d%H +--R ++ a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.508~~~~~$\displaystyle +\int{x^m\csc^{-1}\frac{x}{a}}~dx$} +$$\int{x^m\csc^{-1}\frac{x}{a}}= +\left\{ +\begin{array}{l} +\displaystyle +\frac{x^{m+1}\csc^{-1}(x/a)}{m+1}+\frac{a}{m+1}\int{\frac{x^m}{\sqrt{x^2-a^2}}} +{\rm \ if\ }0 < \csc^{-1}\frac{x}{a} < \frac{\pi}{2}\\ +\\ +\displaystyle +\frac{x^{m+1}\csc^{-1}(x/a)}{m+1}-\frac{a}{m+1}\int{\frac{x^m}{\sqrt{x^2-a^2}}} +{\rm \ if\ }-\frac{\pi}{2} < \csc^{-1}\frac{x}{a} < 0\\ +\end{array} +\right. +$$ +<<*>>= +)clear all + +--S 38 of 38 +aa:=integrate(x^m*acsc(x/a),x) +--R +--R +--R x +--I ++ %H m +--I (1) | acsc(--)%H d%H +--R ++ 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 pp82-84 +\end{thebibliography} +\end{document}