diff --git a/changelog b/changelog index 6f6fcb0..102ba8b 100644 --- a/changelog +++ b/changelog @@ -1,3 +1,7 @@ +20080402 tpd src/input/Makefile add integration regression testing +20080402 tpd src/input/schaum17.input integrals of sin(ax) +20080402 tpd src/input/schaum16.input integrals of x^n \pm a^n +20080402 tpd src/input/schaum15.input integrals of x^4 \pm a^4 20080401 tpd src/input/Makefile add integration regression testing 20080401 tpd src/input/schaum14.input integrals of x^3+a^3 20080401 tpd src/input/schaum13.input integrals of sqrt(ax^2+bx+c) diff --git a/src/input/Makefile.pamphlet b/src/input/Makefile.pamphlet index 2f109e9..6de47f3 100644 --- a/src/input/Makefile.pamphlet +++ b/src/input/Makefile.pamphlet @@ -358,7 +358,8 @@ REGRES= algaggr.regress algbrbf.regress algfacob.regress alist.regress \ schaum1.regress schaum2.regress schaum3.regress schaum4.regress \ schaum5.regress schaum6.regress schaum7.regress schaum8.regress \ schaum9.regress schaum10.regress schaum11.regress schaum12.regress \ - schaum13.regress schaum14.regress \ + schaum13.regress schaum14.regress schaum15.regress schaum16.regress \ + schaum17.regress \ scherk.regress scope.regress seccsc.regress \ segbind.regress seg.regress \ series2.regress series.regress sersolve.regress set.regress \ @@ -635,7 +636,8 @@ FILES= ${OUT}/algaggr.input ${OUT}/algbrbf.input ${OUT}/algfacob.input \ ${OUT}/schaum5.input ${OUT}/schaum6.input ${OUT}/schaum7.input \ ${OUT}/schaum8.input ${OUT}/schaum9.input ${OUT}/schaum10.input \ ${OUT}/schaum11.input ${OUT}/schaum12.input ${OUT}/schaum13.input \ - ${OUT}/schaum14.input \ + ${OUT}/schaum14.input ${OUT}/schaum15.input ${OUT}/schaum16.input \ + ${OUT}/schaum17.input \ ${OUT}/saddle.input \ ${OUT}/scherk.input ${OUT}/scope.input ${OUT}/seccsc.input \ ${OUT}/segbind.input ${OUT}/seg.input ${OUT}/series2.input \ @@ -941,6 +943,8 @@ DOCFILES= \ ${DOC}/schaum9.input.dvi ${DOC}/schaum10.input.dvi \ ${DOC}/schaum11.input.dvi ${DOC}/schaum12.input.dvi \ ${DOC}/schaum13.input.dvi ${DOC}/schaum14.input.dvi \ + ${DOC}/schaum15.input.dvi ${DOC}/schaum16.input.dvi \ + ${DOC}/schaum17.input.dvi \ ${DOC}/s01eaf.input.dvi ${DOC}/s13aaf.input.dvi \ ${DOC}/s13acf.input.dvi ${DOC}/s13adf.input.dvi \ ${DOC}/s14aaf.input.dvi ${DOC}/s14abf.input.dvi \ diff --git a/src/input/schaum15.input.pamphlet b/src/input/schaum15.input.pamphlet new file mode 100644 index 0000000..51a6094 --- /dev/null +++ b/src/input/schaum15.input.pamphlet @@ -0,0 +1,409 @@ +\documentclass{article} +\usepackage{axiom} +\begin{document} +\title{\$SPAD/input schaum15.input} +\author{Timothy Daly} +\maketitle +\eject +\tableofcontents +\eject +\section{\cite{1}:14.311~~~~~$\displaystyle +\int{\frac{dx}{x^4+a^4}}$} +$$\int{\frac{1}{x^4+a^4}}= +\frac{1}{4a^3\sqrt{2}} +\ln\left(\frac{x^2+ax\sqrt{2}+a^2}{x^2-ax\sqrt{2}+a^2}\right) +-\frac{1}{2a^3\sqrt{2}}\tan^{-1}\frac{ax\sqrt{2}}{x^2-a^2} +$$ +<<*>>= +)spool schaum15.output +)set message test on +)set message auto off +)clear all + +--S 1 of 14 +aa:=integrate(1/(x^4+a^4),x) +--R +--R +--R (1) +--R +------+ +------+2 +------+ +--R | 1 8 | 1 4 +-+ | 1 2 +--R |------ log(16a |------ + 4a x\|2 |------ + x ) +--R 4| 12 4| 12 4| 12 +--R \|256a \|256a \|256a +--R + +--R +------+ +------+2 +------+ +--R | 1 8 | 1 4 +-+ | 1 2 +--R - |------ log(16a |------ - 4a x\|2 |------ + x ) +--R 4| 12 4| 12 4| 12 +--R \|256a \|256a \|256a +--R + +--R +------+ +------+ +--R 4 | 1 4 | 1 +--R 4a |------ 4a |------ +--R +------+ 4| 12 +------+ 4| 12 +--R | 1 \|256a | 1 \|256a +--R 2 |------ atan(-------------------- - 2 |------ atan(--------------------) +--R 4| 12 +------+ 4| 12 +------+ +--R \|256a 4 | 1 +-+ \|256a 4 | 1 +-+ +--R 4a |------ - x\|2 4a |------ + x\|2 +--R 4| 12 4| 12 +--R \|256a \|256a +--R / +--R +-+ +--R \|2 +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.312~~~~~$\displaystyle +\int{\frac{x~dx}{x^4+a^4}}$} +$$\int{\frac{x}{x^4+a^4}}= +\frac{1}{2a^2}\tan^{-1}\frac{x^2}{a^2} +$$ +<<*>>= +)clear all + +--S 2 of 14 +aa:=integrate(x/(x^4+a^4),x) +--R +--R +--R 2 +--R x +--R atan(--) +--R 2 +--R a +--R (1) -------- +--R 2 +--R 2a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.313~~~~~$\displaystyle +\int{\frac{x^2~dx}{x^4+a^4}}$} +$$\int{\frac{x^2}{x^4+a^4}}= +\frac{1}{4a\sqrt{2}} +\ln\left(\frac{x^2-ax\sqrt{2}+a^2}{x^2+ax\sqrt{2}+a^2}\right) +-\frac{1}{2a\sqrt{2}}\tan^{-1}\frac{ax\sqrt{2}}{x^2-a^2} +$$ +<<*>>= +)clear all + +--S 3 of 14 +aa:=integrate(x^2/(x^4+a^4),x) +--R +--R +--R (1) +--R +-----+ +-----+3 +-----+2 +--R | 1 4 +-+ | 1 4 | 1 2 +--R - |----- log(64a x\|2 |----- + 16a |----- + x ) +--R 4| 4 4| 4 4| 4 +--R \|256a \|256a \|256a +--R + +--R +-----+ +-----+3 +-----+2 +--R | 1 4 +-+ | 1 4 | 1 2 +--R |----- log(- 64a x\|2 |----- + 16a |----- + x ) +--R 4| 4 4| 4 4| 4 +--R \|256a \|256a \|256a +--R + +--R +-----+3 +-----+3 +--R 4 | 1 4 | 1 +--R 64a |----- 64a |----- +--R +-----+ 4| 4 +-----+ 4| 4 +--R | 1 \|256a | 1 \|256a +--R 2 |----- atan(--------------------- - 2 |----- atan(---------------------) +--R 4| 4 +-----+3 4| 4 +-----+3 +--R \|256a 4 | 1 +-+ \|256a 4 | 1 +-+ +--R 64a |----- - x\|2 64a |----- + x\|2 +--R 4| 4 4| 4 +--R \|256a \|256a +--R / +--R +-+ +--R \|2 +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.314~~~~~$\displaystyle +\int{\frac{x^3~dx}{x^4+a^4}}$} +$$\int{\frac{x^3}{x^4+a^4}}= +\frac{1}{4}\ln(x^4+a^4) +$$ +<<*>>= +)clear all + +--S 4 of 14 +aa:=integrate(x^3/(x^4+a^4),x) +--R +--R +--R 4 4 +--R log(x + a ) +--R (1) ------------ +--R 4 +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.315~~~~~$\displaystyle +\int{\frac{dx}{x(x^4+a^4)}}~dx$} +$$\int{\frac{1}{x(x^4+a^4)}}= +\frac{1}{4a^4}\ln\left(\frac{x^4}{x^4+a^4}\right) +$$ +<<*>>= +)clear all + +--S 5 of 14 +aa:=integrate(1/(x*(x^4+a^4)),x) +--R +--R +--R 4 4 +--R - log(x + a ) + 4log(x) +--R (1) ------------------------ +--R 4 +--R 4a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.316~~~~~$\displaystyle +\int{\frac{dx}{x^2(x^4+a^4)}}$} +$$\int{\frac{1}{x^2(x^4+a^4)}}= +-\frac{1}{a^4x}-\frac{1}{4a^5\sqrt{2}} +\ln\left(\frac{x^2-ax\sqrt{2}+a^2}{x^2+ax\sqrt{2}+a^2}\right) ++\frac{1}{2a^5\sqrt{2}}\tan^{-1}\frac{ax\sqrt{2}}{x^2-a^2} +$$ +<<*>>= +)clear all + +--S 6 of 14 +aa:=integrate(1/(x^2*(x^4+a^4)),x) +--R +--R +--R (1) +--R +------+ +------+3 +------+2 +--R 4 | 1 16 +-+ | 1 12 | 1 2 +--R a x |------ log(64a x\|2 |------ + 16a |------ + x ) +--R 4| 20 4| 20 4| 20 +--R \|256a \|256a \|256a +--R + +--R +------+ +------+3 +------+2 +--R 4 | 1 16 +-+ | 1 12 | 1 2 +--R - a x |------ log(- 64a x\|2 |------ + 16a |------ + x ) +--R 4| 20 4| 20 4| 20 +--R \|256a \|256a \|256a +--R + +--R +------+3 +--R 16 | 1 +--R 64a |------ +--R +------+ 4| 20 +--R 4 | 1 \|256a +--R - 2a x |------ atan(-----------------------) +--R 4| 20 +------+3 +--R \|256a 16 | 1 +-+ +--R 64a |------ - x\|2 +--R 4| 20 +--R \|256a +--R + +--R +------+3 +--R 16 | 1 +--R 64a |------ +--R +------+ 4| 20 +--R 4 | 1 \|256a +-+ +--R 2a x |------ atan(----------------------- - \|2 +--R 4| 20 +------+3 +--R \|256a 16 | 1 +-+ +--R 64a |------ + x\|2 +--R 4| 20 +--R \|256a +--R / +--R 4 +-+ +--R a x\|2 +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.317~~~~~$\displaystyle +\int{\frac{dx}{x^3(x^4+a^4)}}$} +$$\int{\frac{1}{x^3(x^4+a^4)}}= +-\frac{1}{2a^4x^2}-\frac{1}{2a^6}\tan^{-1}\frac{x^2}{a^2} +$$ +<<*>>= +)clear all + +--S 7 of 14 +aa:=integrate(1/(x^3*(x^4+a^4)),x) +--R +--R +--R 2 +--R 2 x 2 +--R - x atan(--) - a +--R 2 +--R a +--R (1) ----------------- +--R 6 2 +--R 2a x +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.318~~~~~$\displaystyle +\int{\frac{dx}{(x^4-a^4)}}$} +$$\int{\frac{1}{(x^4-a^4)}}= +\frac{1}{4a^3}\ln\left(\frac{x-a}{x+a}\right) +-\frac{1}{2a^3}\tan^{-1}\frac{x}{a} +$$ +<<*>>= +)clear all + +--S 8 of 14 +aa:=integrate(1/(x^4-a^4),x) +--R +--R +--R x +--R - log(x + a) + log(x - a) - 2atan(-) +--R a +--R (1) ------------------------------------ +--R 3 +--R 4a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.319~~~~~$\displaystyle +\int{\frac{x~dx}{(x^4-a^4)}}$} +$$\int{\frac{x}{(x^4-a^4)}}= +\frac{1}{4a^2}\ln\left(\frac{x^2-a^2}{x^2+a^2}\right) +$$ +<<*>>= +)clear all + +--S 9 of 14 +aa:=integrate(x/(x^4-a^4),x) +--R +--R +--R 2 2 2 2 +--R - log(x + a ) + log(x - a ) +--R (1) ----------------------------- +--R 2 +--R 4a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.320~~~~~$\displaystyle +\int{\frac{x^2~dx}{x^4-a^4}}$} +$$\int{\frac{x^2}{x^4-a^4}}= +\frac{1}{4a}\ln\left(\frac{x-a}{x+a}\right) ++\frac{1}{2a}\tan^{-1}\frac{x}{a} +$$ +<<*>>= +)clear all + +--S 10 of 14 +aa:=integrate(x^2/(x^4-a^4),x) +--R +--R +--R x +--R - log(x + a) + log(x - a) + 2atan(-) +--R a +--R (1) ------------------------------------ +--R 4a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.321~~~~~$\displaystyle +\int{\frac{x^3~dx}{x^4-a^4}}$} +$$\int{\frac{x^3}{x^4-a^4}}= +\frac{1}{4}\ln(x^4-a^4) +$$ +<<*>>= +)clear all + +--S 11 of 14 +aa:=integrate(x^3/(x^4-a^4),x) +--R +--R +--R 4 4 +--R log(x - a ) +--R (1) ------------ +--R 4 +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.322~~~~~$\displaystyle +\int{\frac{dx}{x(x^4-a^4)}}$} +$$\int{\frac{1}{x(x^4-a^4)}}= +\frac{1}{4a^4}\ln\left(\frac{x^4-a^4}{x^4}\right) +$$ +<<*>>= +)clear all + +--S 12 of 14 +aa:=integrate(1/(x*(x^4-a^4)),x) +--R +--R +--R 4 4 +--R log(x - a ) - 4log(x) +--R (1) ---------------------- +--R 4 +--R 4a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.323~~~~~$\displaystyle +\int{\frac{dx}{x^2(x^4-a^4)}}$} +$$\int{\frac{1}{x^2(x^4-a^4)}}= +\frac{1}{a^4x}+\frac{1}{4a^5}\ln\left(\frac{x-a}{x+a}\right) ++\frac{1}{2a^5}\tan^{-1}\frac{x}{a} +$$ +<<*>>= +)clear all + +--S 13 of 14 +aa:=integrate(1/(x^2*(x^4-a^4)),x) +--R +--R +--R x +--R - x log(x + a) + x log(x - a) + 2x atan(-) + 4a +--R a +--R (1) ----------------------------------------------- +--R 5 +--R 4a x +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.324~~~~~$\displaystyle +\int{\frac{dx}{x^3(x^4-a^4)}}$} +$$\int{\frac{1}{x^3(x^4-a^4)}}= +\frac{1}{2a^4x^2}+\frac{1}{4a^6}\ln\left(\frac{x^2-a^2}{x^2+a^2}\right) +$$ +<<*>>= +)clear all + +--S 14 of 14 +aa:=integrate(1/(x^3*(x^4-a^4)),x) +--R +--R +--R 2 2 2 2 2 2 2 +--R - x log(x + a ) + x log(x - a ) + 2a +--R (1) --------------------------------------- +--R 6 2 +--R 4a 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 pp73-74 +\end{thebibliography} +\end{document} diff --git a/src/input/schaum16.input.pamphlet b/src/input/schaum16.input.pamphlet new file mode 100644 index 0000000..2246f9a --- /dev/null +++ b/src/input/schaum16.input.pamphlet @@ -0,0 +1,394 @@ +\documentclass{article} +\usepackage{axiom} +\begin{document} +\title{\$SPAD/input schaum16.input} +\author{Timothy Daly} +\maketitle +\eject +\tableofcontents +\eject +\section{\cite{1}:14.325~~~~~$\displaystyle +\int{\frac{dx}{x(x^n+a^n)}}$} +$$\int{\frac{1}{x(x^n+a^n)}}= +\frac{1}{na^n}\ln\frac{x^n}{x^n+a^n} +$$ +<<*>>= +)spool schaum16.output +)set message test on +)set message auto off +)clear all + +--S 1 of 14 +aa:=integrate(1/x*(x^n+a^n),x) +--R +--R +--R n log(x) n +--R %e + n log(x)a +--R (1) ----------------------- +--R n +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.326~~~~~$\displaystyle +\int{\frac{x^{n-1}~dx}{x^n+a^n}}$} +$$\int{\frac{x^{n-1}}{x^n+a^n}}= +\frac{1}{n}\ln(x^n+a^n) +$$ +<<*>>= +)clear all + +--S 2 of 14 +aa:=integrate(x^(n-1)/(x^n+a^n),x) +--R +--R +--R n log(x) n +--R log(%e + a ) +--R (1) -------------------- +--R n +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.327~~~~~$\displaystyle +\int{\frac{x^m~dx}{(x^n+a^n)^r}}$} +$$\int{\frac{x^m}{(x^n+a^n)^r}}= +\int{\frac{x^{m-n}}{(x^n+a^n)^{r-1}}} +-a^n\int{\frac{x^{m-n}}{(x^n+a^n)^r}} +$$ +<<*>>= +)clear all + +--S 3 of 14 +aa:=integrate(x^m/(x^n+a^n)^r,x) +--R +--R +--R x m +--R ++ %J +--R (1) | ----------- d%J +--R ++ n n r +--R (a + %J ) +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.328~~~~~$\displaystyle +\int{\frac{dx}{x^m(x^n+a^n)^r}}$} +$$\int{\frac{1}{x^m(x^n+a^n)^r}}= +\frac{1}{a^n}\int{\frac{1}{x^m(x^n+a^n)^{r-1}}} +-\frac{1}{a^n}\int{\frac{1}{x^{m-n}(x^n+a^n)^r}} +$$ +<<*>>= +)clear all + +--S 4 of 14 +aa:=integrate(1/(x^m*(x^n+a^n)^r),x) +--R +--R +--R x +--R ++ 1 +--R (1) | -------------- d%J +--R ++ m n n r +--R %J (a + %J ) +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.329~~~~~$\displaystyle +\int{\frac{dx}{x\sqrt{x^n+a^n}}}$} +$$\int{\frac{1}{x\sqrt{x^n+a^n}}}= +\frac{1}{n\sqrt{a^n}}\ln\left(\frac{\sqrt{x^n+a^n}-\sqrt{a^n}} +{\sqrt{x^n+a^n}+\sqrt{a^n}}\right) +$$ +<<*>>= +)clear all + +--S 5 of 14 +aa:=integrate(1/(x*sqrt(x^n+a^n)),x) +--R +--R +--R (1) +--R +---------------+ +--+ +--R n | n log(x) n n log(x) n | n +--R - 2a \|%e + a + (%e + 2a )\|a +--R log(-------------------------------------------------) +--R n log(x) +--R %e +--R [------------------------------------------------------, +--R +--+ +--R | n +--R n\|a +--R +----+ +---------------+ +--R | n | n log(x) n +--R \|- a \|%e + a +--R 2atan(-------------------------) +--R n +--R a +--R - --------------------------------] +--R +----+ +--R | n +--R n\|- a +--R Type: Union(List Expression Integer,...) +--E +@ + +\section{\cite{1}:14.330~~~~~$\displaystyle +\int{\frac{dx}{x(x^n-a^n)}}$} +$$\int{\frac{1}{x(x^n-a^n)}}= +\frac{1}{na^n}\ln\left(\frac{x^n-a^n}{x^n}\right) +$$ +<<*>>= +)clear all + +--S 6 of 14 +aa:=integrate(1/(x*(x^n-a^n)),x) +--R +--R +--R n log(x) n +--R log(%e - a ) - n log(x) +--R (1) ------------------------------- +--R n +--R n a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.331~~~~~$\displaystyle +\int{\frac{x^{n-1}dx}{x^n-a^n}}$} +$$\int{\frac{x^{n-1}}{x^n-a^n}}= +\frac{1}{n}\ln(x^n-a^n) +$$ +<<*>>= +)clear all + +--S 7 of 14 +aa:=integrate(x^(n-1)/(x^n-a^n),x) +--R +--R +--R n log(x) n +--R log(%e - a ) +--R (1) -------------------- +--R n +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.332~~~~~$\displaystyle +\int{\frac{x^m~dx}{(x^n-a^n)^r}}$} +$$\int{\frac{x^m}{(x^n-a^n)^r}}= +a^n\int{\frac{x^{m-n}}{(x^n-a^n)^r}} ++\int{\frac{x^{m-n}}{(x^n-a^n)^{r-1}}} +$$ +<<*>>= +)clear all + +--S 8 of 14 +aa:=integrate(x^m/(x^n-a^n)^r,x) +--R +--R +--R x m +--R ++ %J +--R (1) | ------------- d%J +--R ++ n n r +--R (- a + %J ) +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.333~~~~~$\displaystyle +\int{\frac{dx}{x^m(x^n-a^n)^r}}$} +$$\int{\frac{1}{x^m(x^n-a^n)^r}}= +\frac{1}{a^n}\int{\frac{1}{x^{m-n}(x^n-a^n)^r}} +-\frac{1}{a^n}\int{\frac{1}{x^m(x^n-a^n)^{r-1}}} +$$ +<<*>>= +)clear all + +--S 9 of 14 +aa:=integrate(1/(x^m*(x^n-a^n)^r),x) +--R +--R +--R x +--R ++ 1 +--R (1) | ---------------- d%J +--R ++ m n n r +--R %J (- a + %J ) +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.334~~~~~$\displaystyle +\int{\frac{dx}{x\sqrt{x^n-a^n}}}$} +$$\int{\frac{1}{x\sqrt{x^n-a^n}}}= +\frac{2}{n\sqrt{a^n}}\cos^{-1}\sqrt{\frac{a^n}{x^n}} +$$ +<<*>>= +)clear all + +--S 10 of 14 +aa:=integrate(1/(x*sqrt(x^n-a^n)),x) +--R +--R +--R (1) +--R +---------------+ +----+ +--R n | n log(x) n n log(x) n | n +--R 2a \|%e - a + (%e - 2a )\|- a +--R log(-------------------------------------------------) +--R n log(x) +--R %e +--R [------------------------------------------------------, +--R +----+ +--R | n +--R n\|- a +--R +--+ +---------------+ +--R | n | n log(x) n +--R \|a \|%e - a +--R 2atan(-----------------------) +--R n +--R a +--R ------------------------------] +--R +--+ +--R | n +--R n\|a +--R Type: Union(List Expression Integer,...) +--E +@ + +\section{\cite{1}:14.335~~~~~$\displaystyle +\int{\frac{x^{p-1}~dx}{x^{2m}+a^{2m}}}$ provided $0
>= +)clear all + +--S 11 of 14 +aa:=integrate(x^(p-1)/(x^(2*m)+a^(2*m)),x) +--R +--R +--R x p - 1 +--R ++ %J +--R (1) | ---------- d%J +--R ++ 2m 2m +--R a + %J +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.336~~~~~$\displaystyle +\int{\frac{x^{p-1}dx}{x^{2m}-a^{2m}}}$ provided $0
>= +)clear all + +--S 12 of 14 +aa:=integrate(x^(p-1)/(x^(2*m)-a^(2*m)),x) +--R +--R +--R x p - 1 +--R ++ %J +--R (1) | - ---------- d%J +--R ++ 2m 2m +--R a - %J +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.337~~~~~$\displaystyle +\int{\frac{x^{p-1}~dx}{x^{2m+1}+a^{2m+1}}}$ provided $0
>= +)clear all + +--S 13 of 14 +aa:=integrate(x^(p-1)/(x^(2*m+1)+a^(2*m+1)),x) +--R +--R +--R x p - 1 +--R ++ %J +--R (1) | ------------------ d%J +--R ++ 2m + 1 2m + 1 +--R a + %J +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.338~~~~~$\displaystyle +\int{\frac{x^{p-1}~dx}{x^{2m+1}-a^{2m+1}}}$ provided $0
>= +)clear all + +--S 14 of 14 +aa:=integrate(x^(p-1)/(x^(2*m+1)-a^(2*m+1)),x) +--R +--R +--R x p - 1 +--R ++ %J +--R (1) | - ------------------ d%J +--R ++ 2m + 1 2m + 1 +--R a - %J +--R Type: Union(Expression Integer,...) +--E + +)spool +)lisp (bye) +@ + +\eject +\begin{thebibliography}{99} +\bibitem{1} Spiegel, Murray R. +{\sl Mathematical Handbook of Formulas and Tables}\\ +Schaum's Outline Series McGraw-Hill 1968 pp74-75 +\end{thebibliography} +\end{document} diff --git a/src/input/schaum17.input.pamphlet b/src/input/schaum17.input.pamphlet new file mode 100644 index 0000000..7ab11d7 --- /dev/null +++ b/src/input/schaum17.input.pamphlet @@ -0,0 +1,779 @@ +\documentclass{article} +\usepackage{axiom} +\begin{document} +\title{\$SPAD/input schaum17.input} +\author{Timothy Daly} +\maketitle +\eject +\tableofcontents +\eject +\section{\cite{1}:14.339~~~~~$\displaystyle +\int{\sin ax ~dx}$} +$$\int{\sin ax}= +-\frac{\cos{ax}}{a} +$$ +<<*>>= +)spool schaum17.output +)set message test on +)set message auto off +)clear all + +--S 1 of 30 +aa:=integrate(sin(a*x),x) +--R +--R +--R cos(a x) +--R (1) - -------- +--R a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.340~~~~~$\displaystyle +\int{x\sin{ax}~dx}$} +$$\int{x\sin{ax}}= +\frac{sin{ax}}{a^2}-\frac{x\cos{ax}}{a} +$$ +<<*>>= +)clear all + +--S 2 of 30 +aa:=integrate(x*sin(a*x),x) +--R +--R +--R sin(a x) - a x cos(a x) +--R (1) ----------------------- +--R 2 +--R a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.341~~~~~$\displaystyle +\int{x^2\sin{ax}~dx}$} +$$\int{x^2\sin{ax}}= +\frac{2x}{a^2}\sin{ax}+\left(\frac{2}{a^3}-\frac{x^2}{a}\right)\cos{ax} +$$ +<<*>>= +)clear all + +--S 3 of 30 +aa:=integrate(x^2*sin(a*x),x) +--R +--R +--R 2 2 +--R 2a x sin(a x) + (- a x + 2)cos(a x) +--R (1) ------------------------------------ +--R 3 +--R a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.342~~~~~$\displaystyle +\int{x^3\sin{ax}~dx}$} +$$\int{x^3\sin{ax}}= +\left(\frac{3x^2}{a^2}-\frac{6}{a^4}\right)\sin{ax} ++\left(\frac{6x}{a^3}-\frac{x^3}{a}\right)\cos{ax} +$$ +<<*>>= +)clear all + +--S 4 of 30 +aa:=integrate(x^3*sin(a*x),x) +--R +--R +--R 2 2 3 3 +--R (3a x - 6)sin(a x) + (- a x + 6a x)cos(a x) +--R (1) --------------------------------------------- +--R 4 +--R a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.343~~~~~$\displaystyle +\int{\frac{\sin{ax}}{x}}~dx$} +$$\int{\frac{\sin{ax}}{x}}= +ax-\frac{(ax)^3}{3\cdot 3!}+\frac{(ax)^5}{5\cdot 5!}-\cdots +$$ +<<*>>= +)clear all + +--S 5 of 30 +aa:=integrate(sin(x)/x,x) +--R +--R +--R (1) Si(x) +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.344~~~~~$\displaystyle +\int{\frac{\sin{ax}}{x^2}}~dx$} +$$\int{\frac{\sin{ax}}{x^2}}= +-\frac{\sin{ax}}{x}+a\int{\frac{\cos{ax}}{x}} +$$ +<<*>>= +)clear all + +--S 6 of 30 +aa:=integrate(sin(a*x)/x^2,x) +--R +--R +--R x +--R ++ sin(%I a) +--R (1) | --------- d%I +--R ++ 2 +--R %I +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.345~~~~~$\displaystyle +\int{\frac{dx}{\sin{ax}}}$} +$$\int{\frac{1}{\sin{ax}}}= +\frac{1}{a}\ln(\csc{ax}-\cot{ax})= +\frac{1}{a}\ln\tan\frac{ax}{2} +$$ +<<*>>= +)clear all + +--S 7 of 30 +aa:=integrate(1/sin(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}:14.346~~~~~$\displaystyle +\int{\frac{x~dx}{\sin{ax}}}$} +$$\int{\frac{x}{\sin{ax}}}= +\frac{1}{a^2}\left\{ +ax+\frac{(ax)^3}{18}+\frac{7(ax)^5}{1800}+\cdots+ +\frac{2(2^{2n-1}-1)B_n(ax)^{2n+1}}{(2n+1)!}+\cdots\right\} +$$ +<<*>>= +)clear all + +--S 8 of 30 +aa:=integrate(x/sin(a*x),x) +--R +--R +--R x +--R ++ %I +--R (1) | --------- d%I +--R ++ sin(%I a) +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.347~~~~~$\displaystyle +\int{\sin^2{ax}}~dx$} +$$\int{\sin^2{ax}}= +\frac{x}{2}-\frac{\sin{2ax}}{4a} +$$ +<<*>>= +)clear all + +--S 9 of 30 +aa:=integrate(sin(a*x)^2,x) +--R +--R +--R - cos(a x)sin(a x) + a x +--R (1) ------------------------ +--R 2a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.348~~~~~$\displaystyle +\int{x\sin^2{ax}}~dx$} +$$\int{x\sin^2{ax}}= +\frac{x^2}{4}-\frac{x\sin{2ax}}{4a}-\frac{\cos{2ax}}{8a^2} +$$ +<<*>>= +)clear all + +--S 10 of 30 +aa:=integrate(x*sin(a*x)^2,x) +--R +--R +--R 2 2 2 +--R - 2a x cos(a x)sin(a x) - cos(a x) + a x +--R (1) ------------------------------------------ +--R 2 +--R 4a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.349~~~~~$\displaystyle +\int{\sin^3{ax}}~dx$} +$$\int{\sin^3{ax}}= +-\frac{\cos{ax}}{a}+\frac{\cos^3{ax}}{3a} +$$ +<<*>>= +)clear all + +--S 11 of 30 +aa:=integrate(sin(a*x)^3,x) +--R +--R +--R 3 +--R cos(a x) - 3cos(a x) +--R (1) --------------------- +--R 3a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.350~~~~~$\displaystyle +\int{\sin^4{ax}}~dx$} +$$\int{\sin^4{ax}}= +\frac{3x}{8}-\frac{\sin{2ax}}{4a}+\frac{\sin{4ax}}{32a} +$$ +<<*>>= +)clear all + +--S 12 of 30 +aa:=integrate(sin(a*x)^4,x) +--R +--R +--R 3 +--R (2cos(a x) - 5cos(a x))sin(a x) + 3a x +--R (1) --------------------------------------- +--R 8a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.351~~~~~$\displaystyle +\int{\frac{dx}{\sin^2{ax}}}$} +$$\int{\frac{1}{\sin^2{ax}}}= +-\frac{1}{a}\cot{ax} +$$ +<<*>>= +)clear all + +--S 13 of 30 +aa:=integrate(1/sin(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}:14.352~~~~~$\displaystyle +\int{\frac{dx}{\sin^3{ax}}}$} +$$\int{\frac{1}{\sin^3{ax}}}= +-\frac{\cos{ax}}{2a\sin^2{ax}}+\frac{1}{2a}\ln\tan\frac{ax}{2} +$$ +<<*>>= +)clear all + +--S 14 of 30 +aa:=integrate(1/sin(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}:14.353~~~~~$\displaystyle +\int{\sin{px}\sin{qx}}~dx$} +$$\int{\sin{px}\sin{qx}}= +\frac{\sin{(p-q)x}}{2(p-q)}-\frac{\sin{(p+q)x}}{2(p+q)} +$$ +<<*>>= +)clear all + +--S 15 of 30 +aa:=integrate(sin(p*x)*sin(q*x),x) +--R +--R +--R p cos(p x)sin(q x) - q cos(q x)sin(p x) +--R (1) --------------------------------------- +--R 2 2 +--R q - p +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.354~~~~~$\displaystyle +\int{\frac{dx}{1-\sin{ax}}}$} +$$\int{\frac{1}{1-\sin{ax}}}= +\frac{1}{a}\tan\left(\frac{\pi}{4}+\frac{ax}{2}\right) +$$ +<<*>>= +)clear all + +--S 16 of 30 +aa:=integrate(1/(1-sin(a*x)),x) +--R +--R +--R - 2cos(a x) - 2 +--R (1) --------------------------- +--R a sin(a x) - a cos(a x) - a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.355~~~~~$\displaystyle +\int{\frac{x~dx}{1-\sin{ax}}}$} +$$\int{\frac{x}{1-\sin{ax}}}= +\frac{x}{a}\tan\left(\frac{\pi}{4}+\frac{ax}{2}\right) ++\frac{2}{a^2}\ln~\sin\left(\frac{\pi}{4}-\frac{ax}{2}\right) +$$ +<<*>>= +)clear all + +--S 17 of 30 +aa:=integrate(x/(1-sin(ax)),x) +--R +--R +--R 2 +--R x +--R (1) - ------------ +--R 2sin(ax) - 2 +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.356~~~~~$\displaystyle +\int{\frac{dx}{1+\sin{ax}}}$} +$$\int{\frac{1}{1+\sin{ax}}}= +-\frac{1}{a}\tan\left(\frac{\pi}{4}-\frac{ax}{2}\right) +$$ +<<*>>= +)clear all + +--S 18 of 30 +aa:=integrate(1/(1+sin(ax)),x) +--R +--R +--R x +--R (1) ----------- +--R sin(ax) + 1 +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.357~~~~~$\displaystyle +\int{\frac{x~dx}{1+\sin{ax}}}$} +$$\int{\frac{x}{1+\sin{ax}}}= +-\frac{x}{a}\tan\left(\frac{\pi}{4}-\frac{ax}{2}\right) ++\frac{2}{a^2}\ln~\sin\left(\frac{\pi}{4}+\frac{ax}{2}\right) +$$ +<<*>>= +)clear all + +--S 19 of 30 +aa:=integrate(x/(1+sin(a*x)),x) +--R +--R +--R (1) +--R sin(a x) + cos(a x) + 1 +--R (2sin(a x) + 2cos(a x) + 2)log(-----------------------) +--R cos(a x) + 1 +--R + +--R 2 +--R (- sin(a x) - cos(a x) - 1)log(------------) + a x sin(a x) +--R cos(a x) + 1 +--R + +--R - a x cos(a x) - a x +--R / +--R 2 2 2 +--R a sin(a x) + a cos(a x) + a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.358~~~~~$\displaystyle +\int{\frac{dx}{(1-\sin{ax})^2}}$} +$$\int{\frac{1}{(1-\sin{ax})^2}}= +\frac{1}{2a}\tan\left(\frac{\pi}{4}+\frac{ax}{2}\right) ++\frac{1}{6a}\tan^3\left(\frac{\pi}{4}+\frac{ax}{2}\right) +$$ +<<*>>= +)clear all + +--S 20 of 30 +aa:=integrate(1/(1-sin(a*x))^2,x) +--R +--R +--R 2 +--R (3cos(a x) + 3)sin(a x) + cos(a x) - 4cos(a x) - 5 +--R (1) ------------------------------------------------------------ +--R 2 +--R (3a cos(a x) + 6a)sin(a x) + 3a cos(a x) - 3a cos(a x) - 6a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.359~~~~~$\displaystyle +\int{\frac{dx}{(1+\sin{ax})^2}}$} +$$\int{\frac{1}{(1+\sin{ax})^2}}= +-\frac{1}{2a}\tan\left(\frac{\pi}{4}-\frac{ax}{2}\right) +-\frac{1}{6a}\tan^3\left(\frac{\pi}{4}-\frac{ax}{2}\right) +$$ +<<*>>= +)clear all + +--S 21 of 30 +aa:=integrate(1/(1+sin(a*x))^2,x) +--R +--R +--R 2 +--R (- 3cos(a x) - 3)sin(a x) + cos(a x) - 4cos(a x) - 5 +--R (1) ------------------------------------------------------------ +--R 2 +--R (3a cos(a x) + 6a)sin(a x) - 3a cos(a x) + 3a cos(a x) + 6a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.360~~~~~$\displaystyle +\int{\frac{dx}{p+q\sin{ax}}}$} +$$\int{\frac{1}{p+q\sin{ax}}}= +\left\{ +\begin{array}{l} +\displaystyle +\frac{2}{a\sqrt{p^2-q^q}} +\tan^{-1}\frac{p\tan{\frac{1}{2}ax}+q}{\sqrt{p^2-q^2}}\\ +\\ +\displaystyle +\frac{1}{a\sqrt{q^2-p^2}}\ln\left(\frac{p\tan{\frac{1}{2}ax}+q-\sqrt{q^2-p^2}} +{p\tan{\frac{1}{2}ax}+q+\sqrt{q^2-p^2}}\right) +\end{array} +\right. +$$ +<<*>>= +)clear all + +--S 22 of 30 +aa:=integrate(1/(p+q*sin(a*x)),x) +--R +--R +--R (1) +--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\|q - p +--R , +--R +---------+ +--R | 2 2 +--R (p sin(a x) + q cos(a x) + q)\|- q + p +--R 2atan(-----------------------------------------) +--R 2 2 2 2 +--R (q - p )cos(a x) + q - p +--R - ------------------------------------------------] +--R +---------+ +--R | 2 2 +--R a\|- q + p +--R Type: Union(List Expression Integer,...) +--E +@ + +\section{\cite{1}:14.361~~~~~$\displaystyle +\int{\frac{dx}{(p+q\sin{ax})^2}}$} +$$\int{\frac{1}{(p+q\sin{ax})^2}}= +\frac{q\cos{ax}}{a(p^2-q^2)(p+q\sin{ax})} ++\frac{p}{p^2-q^2}\int{\frac{1}{p+q\sin{ax}}} +$$ +<<*>>= +)clear all + +--S 23 of 30 +aa:=integrate(1/(p+q*sin(a*x))^2,x) +--R +--R +--R (1) +--R [ +--R 2 3 +--R (p q sin(a x) + 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 2 +--R (- q sin(a x) - p q cos(a x) - p q)\|q - p +--R / +--R +-------+ +--R 3 3 2 2 4 | 2 2 +--R ((a p q - a p q)sin(a x) + a p q - a p )\|q - p +--R , +--R +--R +---------+ +--R | 2 2 +--R 2 3 (p sin(a x) + q cos(a x) + q)\|- q + p +--R (2p q sin(a x) + 2p )atan(-----------------------------------------) +--R 2 2 2 2 +--R (q - p )cos(a x) + q - p +--R + +--R +---------+ +--R 2 | 2 2 +--R (- q sin(a x) - p q cos(a x) - p q)\|- q + p +--R / +--R +---------+ +--R 3 3 2 2 4 | 2 2 +--R ((a p q - a p q)sin(a x) + a p q - a p )\|- q + p +--R ] +--R Type: Union(List Expression Integer,...) +--E +@ + +\section{\cite{1}:14.362~~~~~$\displaystyle +\int{\frac{dx}{p^2+q^2\sin^2{ax}}}$} +$$\int{\frac{1}{p^2+q^2\sin^2{ax}}}= +\frac{1}{ap\sqrt{p^2+q^2}}\tan^{-1}\frac{\sqrt{p^2+q^2}\tan{ax}}{p} +$$ +<<*>>= +)clear all + +--S 24 of 30 +aa:=integrate(1/(p^2+a^2*sin(a*x)),x) +--R +--R +--R (1) +--R [ +--R log +--R +---------+ +--R 2 2 4 4 4 | 4 4 +--R (a p sin(a x) + (- p + a )cos(a x) + a )\|- p + a +--R + +--R 6 4 2 2 4 6 2 4 6 +--R (p - a p )sin(a x) + (a p - a )cos(a x) + a p - a +--R / +--R 2 2 +--R a sin(a x) + p +--R / +--R +---------+ +--R | 4 4 +--R a\|- p + a +--R , +--R +-------+ +--R 2 2 2 | 4 4 +--R (p sin(a x) + a cos(a x) + a )\|p - a +--R 2atan(----------------------------------------) +--R 4 4 4 4 +--R (p - a )cos(a x) + p - a +--R -----------------------------------------------] +--R +-------+ +--R | 4 4 +--R a\|p - a +--R Type: Union(List Expression Integer,...) +--E +@ + +\section{\cite{1}:14.363~~~~~$\displaystyle +\int{\frac{dx}{p^2-q^2\sin^2{ax}}}$} +$$\int{\frac{1}{p^2-q^2\sin^2{ax}}}= +\left\{ +\begin{array}{l} +\displaystyle +\frac{1}{ap\sqrt{p^2-q^2}}\tan^{-1}\frac{\sqrt{p^2-q^2}\tan{ax}}{p}\\ +\\ +\displaystyle +\frac{1}{2ap\sqrt{q^2-p^2}}\ln\left(\frac{\sqrt{q^2-p^2}\tan{ax}+p} +{\sqrt{q^2-p^2}\tan{ax}-p}\right) +\end{array} +\right. +$$ +<<*>>= +)clear all + +--S 25 of 30 +aa:=integrate(1/(p^2-q^2*sin(a*x)^2),x) +--R +--R +--R (1) +--R [ +--R log +--R +-------+ +--R 2 2 2 2 2 | 2 2 +--R ((- q + 2p )cos(a x) + q - p )\|q - p +--R + +--R 2 3 +--R (2p q - 2p )cos(a x)sin(a x) +--R / +--R 2 2 2 2 +--R q cos(a x) - q + p +--R / +--R +-------+ +--R | 2 2 +--R 2a p\|q - p +--R , +--R +--R +---------+ +--R | 2 2 +--R p sin(a x)\|- q + p +--R - atan(-------------------------------) +--R 2 2 2 2 +--R (2q - 2p )cos(a x) + 2q - 2p +--R + +--R 2 2 2 2 +--R ((2q - p )cos(a x) + 2q - 2p )sin(a x) +--R - atan(-------------------------------------------) +--R +---------+ +--R 2 | 2 2 +--R (p cos(a x) + 2p cos(a x) + p)\|- q + p +--R / +--R +---------+ +--R | 2 2 +--R a p\|- q + p +--R ] +--R Type: Union(List Expression Integer,...) +--E +@ + +\section{\cite{1}:14.364~~~~~$\displaystyle +\int{x^m\sin{ax}}~dx$} +$$\int{x^m\sin{ax}}= +-\frac{x^m\cos{ax}}{a}+\frac{mx^{m-1}\sin{ax}}{a^2} +-\frac{m(m-1)}{a^2}\int{x^{m-2}\sin{ax}} +$$ +<<*>>= +)clear all + +--S 26 of 30 +aa:=integrate(x^m*sin(a*x),x) +--R +--R +--R x +--R ++ m +--R (1) | sin(%I a)%I d%I +--R ++ +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.365~~~~~$\displaystyle +\int{\frac{\sin{ax}}{x^n}}~dx$} +$$\int{\frac{\sin{ax}}{x^n}}= +-\frac{\sin{ax}}{(n-1)x^{n-1}}+\frac{a}{n-1}\int{\frac{\cos{ax}}{x^{n-1}}} +$$ +<<*>>= +)clear all + +--S 26 of 30 +aa:=integrate(sin(a*x)/x^n,x) +--R +--R +--R x +--R ++ sin(%I a) +--R (1) | --------- d%I +--R ++ n +--R %I +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.366~~~~~$\displaystyle +\int{\sin^n{ax}}~dx$} +$$\int{\sin^n{ax}}= +-\frac{\sin^{n-1}{ax}\cos{ax}}{an}+\frac{n-1}{n}\int{\sin^{n-2}{ax}} +$$ +<<*>>= +)clear all + +--S 28 of 30 +aa:=integrate(sin(a*x)^n,x) +--R +--R +--R x +--R ++ n +--R (1) | sin(%I a) d%I +--R ++ +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.367~~~~~$\displaystyle +\int{\frac{1}{\sin^n{ax}}}~dx$} +$$\int{\frac{1}{\sin^n{ax}}}= +\frac{-\cos{ax}}{a(n-1)\sin^{n-1}{ax}} ++\frac{n-2}{n-1}\int{\frac{1}{\sin^{n-2}{ax}}} +$$ +<<*>>= +)clear all + +--S 29 of 30 +aa:=integrate(1/(sin(a*x))^n,x) +--R +--R +--R x +--R ++ 1 +--R (1) | ---------- d%I +--R ++ n +--R sin(%I a) +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.368~~~~~$\displaystyle +\int{\frac{x~dx}{sin^n{ax}}}$} +$$\int{\frac{x}{sin^n{ax}}}= +\frac{-x\cos{ax}}{a(n-1)\sin^{n-1}{ax}} +-\frac{1}{a^2(n-1)(n-2)\sin^{n-2}{ax}} ++\frac{n-2}{n-1}\int{\frac{x}{\sin^{n-2}{ax}}} +$$ +<<*>>= +)clear all + +--S 30 of 30 +aa:=integrate(x/sin(a*x)^n,x) +--R +--R +--R x +--R ++ %I +--R (1) | ---------- d%I +--R ++ n +--R sin(%I 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 pp75-76 +\end{thebibliography} +\end{document}