diff --git a/changelog b/changelog index 98ee5e5..509310d 100644 --- a/changelog +++ b/changelog @@ -1,3 +1,5 @@ +20080406 tpd src/input/Makefile add integration regression testing +20080406 tpd src/input/schaum19.input integrals of sin(ax) and cos(ax) 20080404 tpd faq FAQ 46: Axiom Trademark information 20080404 tpd faq FAQ 45: Axiom Copyright information 20080403 tpd src/input/Makefile add integration regression testing diff --git a/src/input/Makefile.pamphlet b/src/input/Makefile.pamphlet index dbeb3d7..8170a48 100644 --- a/src/input/Makefile.pamphlet +++ b/src/input/Makefile.pamphlet @@ -359,7 +359,7 @@ 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 \ + schaum17.regress schaum18.regress schaum19.regress \ scherk.regress scope.regress seccsc.regress \ segbind.regress seg.regress \ series2.regress series.regress sersolve.regress set.regress \ @@ -637,7 +637,7 @@ FILES= ${OUT}/algaggr.input ${OUT}/algbrbf.input ${OUT}/algfacob.input \ ${OUT}/schaum8.input ${OUT}/schaum9.input ${OUT}/schaum10.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}/schaum17.input ${OUT}/schaum18.input ${OUT}/schaum19.input \ ${OUT}/saddle.input \ ${OUT}/scherk.input ${OUT}/scope.input ${OUT}/seccsc.input \ ${OUT}/segbind.input ${OUT}/seg.input ${OUT}/series2.input \ @@ -945,6 +945,7 @@ 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}/s01eaf.input.dvi ${DOC}/s13aaf.input.dvi \ ${DOC}/s13acf.input.dvi ${DOC}/s13adf.input.dvi \ ${DOC}/s14aaf.input.dvi ${DOC}/s14abf.input.dvi \ diff --git a/src/input/schaum19.input.pamphlet b/src/input/schaum19.input.pamphlet new file mode 100644 index 0000000..37e89ee --- /dev/null +++ b/src/input/schaum19.input.pamphlet @@ -0,0 +1,888 @@ +\documentclass{article} +\usepackage{axiom} +\begin{document} +\title{\$SPAD/input schaum19.input} +\author{Timothy Daly} +\maketitle +\eject +\tableofcontents +\eject +\section{\cite{1}:14.399~~~~~$\displaystyle +\int{\sin{ax}\cos{ax}}~dx$} +$$\int{\sin{ax}\cos{ax}}= +\frac{\sin^2{ax}}{2a} +$$ +<<*>>= +)spool schaum19.output +)set message test on +)set message auto off +)clear all + +--S 1 of 36 +aa:=integrate(sin(a*x)*cos(a*x),x) +--R +--R +--R 2 +--R cos(a x) +--R (1) - --------- +--R 2a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.400~~~~~$\displaystyle +\int{\sin{px}\cos{qx}}~dx$} +$$\int{\sin{px}\cos{qx}}= +-\frac{cos(p-q)x}{2(p-q)}-\frac{cos(p+q)x}{2(p+q)} +$$ +<<*>>= +)clear all + +--S 2 of 36 +aa:=integrate(sin(p*x)*cos(q*x),x) +--R +--R +--R q sin(p x)sin(q x) + p cos(p x)cos(q x) +--R (1) --------------------------------------- +--R 2 2 +--R q - p +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.401~~~~~$\displaystyle +\int{\sin^n{ax}\cos{ax}}~dx$ provided $n \ne -1$} +$$\int{\sin^n{ax}\cos{ax}}= +\frac{\sin^{n+1}{ax}}{(n+1)a} +$$ +<<*>>= +)clear all + +--S 3 of 36 +aa:=integrate(sin(a*x)^n*cos(a*x),x) +--R +--R +--R n log(sin(a x)) +--R sin(a x)%e +--R (1) ------------------------- +--R a n + a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.402~~~~~$\displaystyle +\int{\cos^n{ax}*sin{ax}}~dx$ provided $n \ne -1$} +$$\int{\cos^n{ax}*sin{ax}}= +-\frac{\cos^{n+1}{ax}}{(n+1)a} +$$ +<<*>>= +)clear all + +--S 4 of 36 +aa:=integrate(cos(a*x)^n*sin(a*x),x) +--R +--R +--R n log(cos(a x)) +--R cos(a x)%e +--R (1) - ------------------------- +--R a n + a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.403~~~~~$\displaystyle +\int{\sin^2{ax}\cos^2{ax}}$} +$$\int{\sin^2{ax}\cos^2{ax}}= +\frac{x}{8}-\frac{\sin{4ax}}{32a} +$$ +<<*>>= +)clear all + +--S 5 of 36 +aa:=integrate(sin(a*x)^2*cos(a*x)^2,x) +--R +--R +--R 3 +--R (- 2cos(a x) + cos(a x))sin(a x) + a x +--R (1) --------------------------------------- +--R 8a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.404~~~~~$\displaystyle +\int{\frac{dx}{\sin{ax}\cos{ax}}}$} +$$\int{\frac{1}{\sin{ax}\cos{ax}}}= +\frac{1}{a}\ln~\tan{ax} +$$ +<<*>>= +)clear all + +--S 6 of 36 +aa:=integrate(1/(sin(a*x)*cos(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}:14.405~~~~~$\displaystyle +\int{\frac{dx}{\sin^2{ax}\cos{ax}}}$} +$$\int{\frac{1}{\sin^2{ax}\cos{ax}}}= +\frac{1}{a}\ln~\tan\left(\frac{\pi}{4}+\frac{ax}{2}\right)-\frac{1}{a\sin{ax}} +$$ +<<*>>= +)clear all + +--S 7 of 36 +aa:=integrate(1/(sin(a*x)^2*cos(a*x)),x) +--R +--R +--R (1) +--R sin(a x) + cos(a x) + 1 +--R sin(a x)log(-----------------------) +--R cos(a x) + 1 +--R + +--R sin(a x) - cos(a x) - 1 +--R - sin(a x)log(-----------------------) - 1 +--R cos(a x) + 1 +--R / +--R a sin(a x) +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.406~~~~~$\displaystyle +\int{\frac{dx}{\sin{ax}\cos^2{ax}}}$} +$$\int{\frac{1}{\sin{ax}\cos^2{ax}}}= +\frac{1}{a}\ln~\tan\frac{ax}{2}+\frac{1}{a\cos{ax}} +$$ +<<*>>= +)clear all + +--S 8 of 36 +aa:=integrate(1/(sin(a*x)*cos(a*x)^2),x) +--R +--R +--R sin(a x) +--R cos(a x)log(------------) + cos(a x) + 1 +--R cos(a x) + 1 +--R (1) ---------------------------------------- +--R a cos(a x) +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.407~~~~~$\displaystyle +\int{\frac{dx}{\sin^2{ax}\cos^2{ax}}}$} +$$\int{\frac{1}{\sin^2{ax}\cos^2{ax}}}= +-\frac{2\cot{2ax}}{a} +$$ +<<*>>= +)clear all + +--S 9 of 36 +aa:=integrate(1/(sin(a*x)^2*cos(a*x)^2),x) +--R +--R +--R 2 +--R - 2cos(a x) + 1 +--R (1) ------------------ +--R a cos(a x)sin(a x) +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.408~~~~~$\displaystyle +\int{\frac{\sin^2{ax}}{\cos{ax}}}~dx$} +$$\int{\frac{\sin^2{ax}}{\cos{ax}}}= +-\frac{\sin{ax}}{a}+\frac{1}{a}\ln~\tan\left(\frac{ax}{2}+\frac{\pi}{4}\right) +$$ +<<*>>= +)clear all + +--S 10 of 36 +aa:=integrate(sin(a*x)^2/cos(a*x),x) +--R +--R +--R sin(a x) + cos(a x) + 1 sin(a x) - cos(a x) - 1 +--R log(-----------------------) - log(-----------------------) - sin(a x) +--R cos(a x) + 1 cos(a x) + 1 +--R (1) ---------------------------------------------------------------------- +--R a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.409~~~~~$\displaystyle +\int{\frac{\cos^2{ax}}{\sin{ax}}}~dx$} +$$\int{\frac{\cos^2{ax}}{\sin{ax}}}= +\frac{\cos{ax}}{a}+\frac{1}{a}\ln~\tan{\frac{ax}{2}} +$$ +<<*>>= +)clear all + +--S 11 of 36 +aa:=integrate(cos(a*x)^2/sin(a*x),x) +--R +--R +--R sin(a x) +--R log(------------) + cos(a x) +--R cos(a x) + 1 +--R (1) ---------------------------- +--R a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.410~~~~~$\displaystyle +\int{\frac{dx}{\cos{ax}(1\pm\sin{ax})}}$} +$$\int{\frac{1}{\cos{ax}(1\pm\sin{ax})}}= +\mp\frac{1}{2a(1\pm\sin{ax})} ++\frac{1}{2a}\ln~\tan\left(\frac{ax}{2}+\frac{\pi}{4}\right) +$$ +<<*>>= +)clear all + +--S 12 of 36 +aa:=integrate(1/(cos(a*x)*(1+sin(a*x))),x) +--R +--R +--R (1) +--R sin(a x) + cos(a x) + 1 +--R (sin(a x) + 1)log(-----------------------) +--R cos(a x) + 1 +--R + +--R sin(a x) - cos(a x) - 1 +--R (- sin(a x) - 1)log(-----------------------) + sin(a x) +--R cos(a x) + 1 +--R / +--R 2a sin(a x) + 2a +--R Type: Union(Expression Integer,...) +--E + +)clear all + +--S 13 of 36 +aa:=integrate(1/(cos(a*x)*(1-sin(a*x))),x) +--R +--R +--R (1) +--R sin(a x) + cos(a x) + 1 +--R (sin(a x) - 1)log(-----------------------) +--R cos(a x) + 1 +--R + +--R sin(a x) - cos(a x) - 1 +--R (- sin(a x) + 1)log(-----------------------) - sin(a x) +--R cos(a x) + 1 +--R / +--R 2a sin(a x) - 2a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.411~~~~~$\displaystyle +\int{\frac{dx}{\sin{ax}(1\pm\cos{ax})}}$} +$$\int{\frac{1}{\sin{ax}(1\pm\cos{ax})}}= +\pm\frac{1}{2a(1\pm\cos{ax})}+\frac{1}{2a}\ln~\tan\frac{ax}{2} +$$ +<<*>>= +)clear all + +--S 14 of 36 +aa:=integrate(1/(sin(a*x)*(1+cos(a*x))),x) +--R +--R +--R sin(a x) +--R (2cos(a x) + 2)log(------------) - cos(a x) + 1 +--R cos(a x) + 1 +--R (1) ----------------------------------------------- +--R 4a cos(a x) + 4a +--R Type: Union(Expression Integer,...) +--E + +)clear all + +--S 15 of 36 +aa:=integrate(1/(sin(a*x)*(1-cos(a*x))),x) +--R +--R +--R sin(a x) +--R (2cos(a x) - 2)log(------------) + cos(a x) + 1 +--R cos(a x) + 1 +--R (1) ----------------------------------------------- +--R 4a cos(a x) - 4a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.412~~~~~$\displaystyle +\int{\frac{dx}{\sin{ax}\pm\cos{ax}}}$} +$$\int{\frac{1}{\sin{ax}\pm\cos{ax}}}= +\frac{1}{a\sqrt{2}}\ln~\tan\left(\frac{ax}{2}\pm\frac{\pi}{8}\right) +$$ +<<*>>= +)clear all + +--S 16 of 36 +aa:=integrate(1/(sin(a*x)+cos(a*x)),x) +--R +--R +--R +-+ +-+ +-+ +--R +-+ (- \|2 + 1)sin(a x) + (\|2 - 1)cos(a x) + \|2 - 2 +--R \|2 log(----------------------------------------------------) +--R sin(a x) + cos(a x) +--R (1) ------------------------------------------------------------- +--R 2a +--R Type: Union(Expression Integer,...) +--E + +)clear all + +--S 17 of 36 +aa:=integrate(1/(sin(a*x)-cos(a*x)),x) +--R +--R +--R +-+ +-+ +-+ +--R +-+ (- \|2 + 1)sin(a x) + (- \|2 + 1)cos(a x) - \|2 + 2 +--R \|2 log(------------------------------------------------------) +--R sin(a x) - cos(a x) +--R (1) --------------------------------------------------------------- +--R 2a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.413~~~~~$\displaystyle +\int{\frac{\sin{ax}~dx}{\sin{ax}\pm\cos{ax}}}$} +$$\int{\frac{\sin{ax}}{\sin{ax}\pm\cos{ax}}}= +\frac{x}{2}\mp\frac{1}{2a}\ln(\sin{ax}\pm\cos{ax}) +$$ +<<*>>= +)clear all + +--S 18 of 36 +aa:=integrate(sin(a*x)/(sin(a*x)+cos(a*x)),x) +--R +--R +--R 2 - 2sin(a x) - 2cos(a x) +--R log(------------) - log(-----------------------) + a x +--R cos(a x) + 1 cos(a x) + 1 +--R (1) ------------------------------------------------------ +--R 2a +--R Type: Union(Expression Integer,...) +--E + +)clear all + +--S 19 of 36 +aa:=integrate(sin(a*x)/(sin(a*x)-cos(a*x)),x) +--R +--R +--R 2sin(a x) - 2cos(a x) 2 +--R log(---------------------) - log(------------) + a x +--R cos(a x) + 1 cos(a x) + 1 +--R (1) ---------------------------------------------------- +--R 2a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.414~~~~~$\displaystyle +\int{\frac{cos{ax}~dx}{\sin{ax}\pm{\cos{ax}}}}$} +$$\int{\frac{cos{ax}}{\sin{ax}\pm{\cos{ax}}}}= +\pm\frac{x}{2}+\frac{1}{2a}\ln(sin{ax}\pm\cos{ax}) +$$ +<<*>>= +)clear all + +--S 20 of 36 +aa:=integrate(cos(a*x)/(sin(a*x)+cos(a*x)),x) +--R +--R +--R 2 - 2sin(a x) - 2cos(a x) +--R - log(------------) + log(-----------------------) + a x +--R cos(a x) + 1 cos(a x) + 1 +--R (1) -------------------------------------------------------- +--R 2a +--R Type: Union(Expression Integer,...) +--E + +)clear all + +--S 21 of 36 +aa:=integrate(cos(a*x)/(sin(a*x)-cos(a*x)),x) +--R +--R +--R 2sin(a x) - 2cos(a x) 2 +--R log(---------------------) - log(------------) - a x +--R cos(a x) + 1 cos(a x) + 1 +--R (1) ---------------------------------------------------- +--R 2a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.415~~~~~$\displaystyle +\int{\frac{\sin{ax}~dx}{p+q\cos{ax}}}$} +$$\int{\frac{\sin{ax}}{p+q\cos{ax}}}= +-\frac{1}{aq}\ln(p+q\cos{ax}) +$$ +<<*>>= +)clear all + +--S 22 of 36 +aa:=integrate(sin(a*x)/(p+q*cos(a*x)),x) +--R +--R +--R 2 - 2q cos(a x) - 2p +--R log(------------) - log(------------------) +--R cos(a x) + 1 cos(a x) + 1 +--R (1) ------------------------------------------- +--R a q +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.416~~~~~$\displaystyle +\int{\frac{\cos{ax}~dx}{p+q\sin{ax}}}$} +$$\int{\frac{\cos{ax}}{p+q\sin{ax}}}= +\frac{1}{aq}\ln(p+q\sin{ax}) +$$ +<<*>>= +)clear all + +--S 23 of 36 +aa:=integrate(cos(a*x)/(p+q*sin(a*x)),x) +--R +--R +--R 2q sin(a x) + 2p 2 +--R log(----------------) - log(------------) +--R cos(a x) + 1 cos(a x) + 1 +--R (1) ----------------------------------------- +--R a q +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.417~~~~~$\displaystyle +\int{\frac{\sin{ax}~dx}{(p+q\cos{ax})^n}}$} +$$\int{\frac{\sin{ax}}{(p+q\cos{ax})^n}}= +\frac{1}{aq(n-1)(p+q\cos{ax})^{n-1}} +$$ +<<*>>= +)clear all + +--S 24 of 36 +aa:=integrate(sin(a*x)/(p+q*cos(a*x))^n,x) +--R +--R +--R q cos(a x) + p +--R (1) ---------------------------------- +--R n log(q cos(a x) + p) +--R (a n - a)q %e +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.418~~~~~$\displaystyle +\int{\frac{\cos{ax}~dx}{(p+q\sin{ax})^n}}$} +$$\int{\frac{\cos{ax}}{(p+q\sin{ax})^n}}= +\frac{-1}{aq(n-1)(p+q\sin{ax})^{n-1}} +$$ +<<*>>= +)clear all + +--S 25 of 36 +aa:=integrate(cos(a*x)/(p+q*sin(a*x))^n,x) +--R +--R +--R - q sin(a x) - p +--R (1) ---------------------------------- +--R n log(q sin(a x) + p) +--R (a n - a)q %e +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.419~~~~~$\displaystyle +\int{\frac{dx}{p\sin{ax}+q\cos{ax}}}$} +$$\int{\frac{1}{p\sin{ax}+q\cos{ax}}}= +\frac{1}{a\sqrt{p^2+q^2}}\ln~\tan\left(\frac{ax+\tan^{-1}(q/p)}{2}\right) +$$ +<<*>>= +)clear all + +--S 26 of 36 +aa:=integrate(1/(p*sin(a*x)+q*cos(a*x)),x) +--R +--R +--R (1) +--R log +--R +-------+ +--R 2 2 2 | 2 2 +--R (p q sin(a x) - p cos(a x) - q - p )\|q + p +--R + +--R 3 2 2 3 2 3 +--R (- q - p q)sin(a x) + (p q + p )cos(a x) + p q + p +--R / +--R p sin(a x) + q cos(a x) +--R / +--R +-------+ +--R | 2 2 +--R a\|q + p +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.420~~~~~$\displaystyle +\int{\frac{dx}{p\sin{ax}+q\cos{ax}+r}}$} +$$\int{\frac{1}{p\sin{ax}+q\cos{ax}+r}}= +\left\{ +\begin{array}{l} +\displaystyle +\frac{2}{a\sqrt{r^2-p^2-q^q}} +\tan^{-1}\left(\frac{p+(r-q)\tan(ax/2)}{\sqrt{r^2-p^2-a^2}}\right)\\ +\\ +\displaystyle +\frac{1}{a\sqrt{p^2+q^2-r^2}}\ln\left( +\frac{p-\sqrt{p^2+q^2-r^2}+(r-q)\tan{(ax/2)}} +{p+\sqrt{p^2+q^2-r^2}+(r-q)\tan{(ax/2)}}\right) +\end{array} +\right. +$$ +<<*>>= +)clear all + +--S 27 of 36 +aa:=integrate(1/(p*sin(a*x)+q*cos(a*x)+r),x) +--R +--R +--R (1) +--R [ +--R log +--R 2 2 2 +--R (p r - p q)sin(a x) + (- r + q r + p )cos(a x) - q r + q +--R + +--R 2 +--R p +--R * +--R +--------------+ +--R | 2 2 2 +--R \|- r + q + p +--R + +--R 3 2 2 2 3 2 +--R (r - q r + (- q - p )r + q + p q)sin(a x) +--R + +--R 2 2 3 2 2 3 +--R (p r - p q - p )cos(a x) + p r - p q - p +--R / +--R p sin(a x) + q cos(a x) + r +--R / +--R +--------------+ +--R | 2 2 2 +--R a\|- r + q + p +--R , +--R +------------+ +--R | 2 2 2 +--R ((r - q)sin(a x) + p cos(a x) + p)\|r - q - p +--R 2atan(-------------------------------------------------) +--R 2 2 2 2 2 2 +--R (r - q - p )cos(a x) + r - q - p +--R --------------------------------------------------------] +--R +------------+ +--R | 2 2 2 +--R a\|r - q - p +--R Type: Union(List Expression Integer,...) +--E +@ + +\section{\cite{1}:14.421~~~~~$\displaystyle +\int{\frac{dx}{p\sin{ax}+q(1+\cos{ax})}}$} +$$\int{\frac{1}{p\sin{ax}+q(1+\cos{ax})}}= +\frac{1}{ap}\ln\left(q+p\tan{\frac{ax}{2}}\right) +$$ +<<*>>= +)clear all + +--S 28 of 36 +aa:=integrate(1/(p*sin(a*x)+q*(1+cos(a*x))),x) +--R +--R +--R p sin(a x) + q cos(a x) + q +--R log(---------------------------) +--R cos(a x) + 1 +--R (1) -------------------------------- +--R a p +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.422~~~~~$\displaystyle +\int{\frac{dx}{p\sin{ax}+q\cos{ax}\pm\sqrt{p^2+q^2}}}$} +$$\int{\frac{1}{p\sin{ax}+q\cos{ax}\pm\sqrt{p^2+q^2}}}= +\frac{-1}{a\sqrt{p^2+q^2}} +\tan\left(\frac{\pi}{4}\mp\frac{ax+\tan^{-1}{(q/p)}}{2}\right) +$$ +<<*>>= +)clear all + +--S 29 of 36 +aa:=integrate(1/(p*sin(a*x)+q*cos(a*x)+sqrt(p^2+q^2)),x) +--R +--R +--R (1) +--R +-------+ +--R 5 2 3 4 5 2 3 4 | 2 2 +--R ((64q + 64p q + 12p q)cos(a x) + 64q + 64p q + 12p q)\|q + p +--R + +--R 6 2 4 4 2 6 6 2 4 4 2 6 +--R (- 64q - 96p q - 36p q - 2p )cos(a x) - 64q - 96p q - 36p q - 2p +--R / +--R 6 2 4 4 2 6 +--R (64a q + 80a p q + 24a p q + a p )sin(a x) +--R + +--R 5 3 3 5 5 3 3 5 +--R (- 32a p q - 32a p q - 6a p q)cos(a x) - 32a p q - 32a p q - 6a p q +--R * +--R +-------+ +--R | 2 2 +--R \|q + p +--R + +--R 7 2 5 4 3 6 +--R (- 64a q - 112a p q - 56a p q - 7a p q)sin(a x) +--R + +--R 6 3 4 5 2 7 6 3 4 +--R (32a p q + 48a p q + 18a p q + a p )cos(a x) + 32a p q + 48a p q +--R + +--R 5 2 7 +--R 18a p q + a p +--R Type: Union(Expression Integer,...) +--E + +)clear all + +--S 30 of 36 +aa:=integrate(1/(p*sin(a*x)+q*cos(a*x)-sqrt(p^2+q^2)),x) +--R +--R +--R (1) +--R +-------+ +--R 5 2 3 4 5 2 3 4 | 2 2 +--R ((64q + 64p q + 12p q)cos(a x) + 64q + 64p q + 12p q)\|q + p +--R + +--R 6 2 4 4 2 6 6 2 4 4 2 6 +--R (64q + 96p q + 36p q + 2p )cos(a x) + 64q + 96p q + 36p q + 2p +--R / +--R 6 2 4 4 2 6 +--R (64a q + 80a p q + 24a p q + a p )sin(a x) +--R + +--R 5 3 3 5 5 3 3 5 +--R (- 32a p q - 32a p q - 6a p q)cos(a x) - 32a p q - 32a p q - 6a p q +--R * +--R +-------+ +--R | 2 2 +--R \|q + p +--R + +--R 7 2 5 4 3 6 +--R (64a q + 112a p q + 56a p q + 7a p q)sin(a x) +--R + +--R 6 3 4 5 2 7 6 3 4 +--R (- 32a p q - 48a p q - 18a p q - a p )cos(a x) - 32a p q - 48a p q +--R + +--R 5 2 7 +--R - 18a p q - a p +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.423~~~~~$\displaystyle +\int{\frac{dx}{p^2\sin^2{ax}+q^2\cos^2{ax}}}$} +$$\int{\frac{1}{p^2\sin^2{ax}+q^2\cos^2{ax}}}= +\frac{1}{apq}\tan^{-1}\left(\frac{p\tan{ax}}{q}\right) +$$ +<<*>>= +)clear all + +--S 31 of 36 +aa:=integrate(1/(p^2*sin(a*x)^2+q^2*cos(a*x)^2),x) +--R +--R +--R 2 2 2 +--R ((q - 2p )cos(a x) - 2p )sin(a x) q sin(a x) +--R - atan(-----------------------------------) + atan(----------------) +--R 2 2p cos(a x) + 2p +--R p q cos(a x) + 2p q cos(a x) + p q +--R (1) -------------------------------------------------------------------- +--R a p q +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.424~~~~~$\displaystyle +\int{\frac{dx}{p^2\sin^2{ax}-q^2\cos^2{ax}}}$} +$$\int{\frac{1}{p^2\sin^2{ax}-q^2\cos^2{ax}}}= +\frac{1}{2apq}\ln\left(\frac{p\tan{ax}-q}{p\tan{ax}+q}\right) +$$ +<<*>>= +)clear all + +--S 32 of 36 +aa:=integrate(1/(p^2*sin(a*x)^2-q^2*cos(a*x)^2),x) +--R +--R +--R 2p sin(a x) - 2q cos(a x) - 2p sin(a x) - 2q cos(a x) +--R log(-------------------------) - log(---------------------------) +--R cos(a x) + 1 cos(a x) + 1 +--R (1) ----------------------------------------------------------------- +--R 2a p q +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.425~~~~~$\displaystyle +\int{\sin^m{ax}\cos^n{ax}}~dx$} +$$\int{\sin^m{ax}\cos^n{ax}}= +\left\{ +\begin{array}{l} +\displaystyle +-\frac{\sin^{m-1}{ax}\cos^{n+1}ax}{a(m+n)} ++\frac{m-1}{m+n}\int{\sin^{m-2}{ax}\cos^n{ax}}\\ +\\ +\displaystyle +\frac{\sin^{m+1}{ax}\cos^{n-1}{ax}}{a(m+n)} ++\frac{n-1}{m+n}\int{\sin^m{ax}\cos^{n-2}{ax}} +\end{array} +\right. +$$ +<<*>>= +)clear all + +--S 33 of 36 +aa:=integrate(sin(a*x)^m*cos(a*x)^n,x) +--R +--R +--R x +--R ++ n m +--I (1) | cos(%H a) sin(%H a) d%H +--R ++ +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.426~~~~~$\displaystyle +\int{\frac{\sin^m{ax}}{\cos^n{ax}}}~dx$} +$$\int{\frac{\sin^m{ax}}{\cos^n{ax}}}= +\left\{ +\begin{array}{l} +\displaystyle +\frac{\sin^{m-1}{ax}}{a(n-1)\cos^{n-1}{ax}} +-\frac{m-1}{n-1}\int{\frac{\sin^{m-2}{ax}}{\cos^{n-2}{ax}}}\\ +\\ +\displaystyle +\frac{\sin^{m+1}{ax}}{a(n-1)\cos^{n-1}{ax}} +-\frac{m-n+2}{n-1}\int{\frac{\sin^m{ax}}{\cos^{n-2}{ax}}}\\ +\\ +\displaystyle +\frac{-\sin^{m-1}{ax}}{a(m-n)\cos^{n-1}{ax}} ++\frac{m-1}{m-n}\int{\frac{\sin^{m-2}{ax}}{\cos^n{ax}}} +\end{array} +\right. +$$ +<<*>>= +)clear all + +--S 34 of 36 +aa:=integrate(sin(a*x)^m/cos(a*x)^n,x) +--R +--R +--R x m +--I ++ sin(%H a) +--I (1) | ---------- d%H +--R ++ n +--I cos(%H a) +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.427~~~~~$\displaystyle +\int{\frac{\cos^m{ax}}{\sin^n{ax}}}~dx$} +$$\int{\frac{\cos^m{ax}}{\sin^n{ax}}}= +\left\{ +\begin{array}{l} +\displaystyle +\frac{-\cos^{m-1}{ax}}{a(n-1)\sin^{n-1}{ax}} +-\frac{m-1}{n-1}\int{\frac{\cos^{m-2}{ax}}{\sin^{n-2}{ax}}}\\ +\\ +\displaystyle +\frac{-\cos^{m+1}{ax}}{a(n-1)\sin^{n-1}{ax}} +-\frac{m-n+2}{n-1}\int{\frac{\cos^m{ax}}{\sin^{n-2}{ax}}}\\ +\\ +\displaystyle +\frac{\cos^{m-1}{ax}}{a(m-n)\sin^{n-1}{ax}} ++\frac{m-1}{m-n}\int{\frac{\cos^{m-2}{ax}}{\sin^n{ax}}} +\end{array} +\right. +$$ +<<*>>= +)clear all + +--S 35 of 36 +aa:=integrate(cos(a*x)^m/sin(a*x)^n,x) +--R +--R +--R x m +--I ++ cos(%H a) +--I (1) | ---------- d%H +--R ++ n +--I sin(%H a) +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.428~~~~~$\displaystyle +\int{\frac{dx}{\sin^m{ax}\cos^n{ax}}}$} +$$\int{\frac{1}{\sin^m{ax}\cos^n{ax}}} +\left\{ +\begin{array}{l} +\displaystyle +\frac{1}{a(n-1)\sin^{m-1}{ax}\cos^{n-1}{ax}} ++\frac{m+n-2}{n-1}\int{\frac{1}{\sin^m{ax}\cos^{n-2}{ax}}}\\ +\\ +\displaystyle +\frac{-1}{a(m-1)\sin^{m-1}{ax}\cos^{n-1}{ax}} ++\frac{m+n-2}{m-1}\int{\frac{1}{\sin^{m-2}{ax}\cos^n{ax}}} +\end{array} +\right. +$$ +<<*>>= +)clear all + +--S 36 of 36 +aa:=integrate(1/(sin(a*x)^m*cos(a*x)^n),x) +--R +--R +--R x +--R ++ 1 +--I (1) | -------------------- d%H +--R ++ n m +--I cos(%H a) sin(%H 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 pp78-80 +\end{thebibliography} +\end{document}