More integrals ======================================================================== diff --git a/changelog b/changelog index 08caf13..bb20757 100644 --- a/changelog +++ b/changelog @@ -1,3 +1,5 @@ +20080409 tpd src/input/Makefile add integration regression testing +20080409 tpd src/input/schaum25.input integrals of e^(a*x) 20080409 tpd readme add Max Tegmark 20080409 tpd src/interp/setq.lisp add Max Tegmark 20080409 tpd src/doc/toe.gif added, by Max Tegmark <[EMAIL PROTECTED]> diff --git a/src/input/Makefile.pamphlet b/src/input/Makefile.pamphlet index c184fbe..9134ab7 100644 --- a/src/input/Makefile.pamphlet +++ b/src/input/Makefile.pamphlet @@ -361,6 +361,7 @@ REGRES= algaggr.regress algbrbf.regress algfacob.regress alist.regress \ schaum13.regress schaum14.regress schaum15.regress schaum16.regress \ schaum17.regress schaum18.regress schaum19.regress schaum20.regress \ schaum21.regress schaum22.regress schaum23.regress schaum24.regress \ + schaum25.regress \ scherk.regress scope.regress seccsc.regress \ segbind.regress seg.regress \ series2.regress series.regress sersolve.regress set.regress \ @@ -640,7 +641,7 @@ FILES= ${OUT}/algaggr.input ${OUT}/algbrbf.input ${OUT}/algfacob.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}/schaum23.input ${OUT}/schaum24.input ${OUT}/schaum25.input \ ${OUT}/saddle.input \ ${OUT}/scherk.input ${OUT}/scope.input ${OUT}/seccsc.input \ ${OUT}/segbind.input ${OUT}/seg.input ${OUT}/series2.input \ @@ -951,6 +952,7 @@ DOCFILES= \ ${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}/schaum25.input.dvi \ ${DOC}/s01eaf.input.dvi ${DOC}/s13aaf.input.dvi \ ${DOC}/s13acf.input.dvi ${DOC}/s13adf.input.dvi \ ${DOC}/s14aaf.input.dvi ${DOC}/s14abf.input.dvi \ diff --git a/src/input/schaum25.input.pamphlet b/src/input/schaum25.input.pamphlet new file mode 100644 index 0000000..250e7c3 --- /dev/null +++ b/src/input/schaum25.input.pamphlet @@ -0,0 +1,383 @@ +\documentclass{article} +\usepackage{axiom} +\begin{document} +\title{\$SPAD/input schaum25.input} +\author{Timothy Daly} +\maketitle +\eject +\tableofcontents +\eject +\section{\cite{1}:14.509~~~~~$\displaystyle +\int{e^{ax}}~dx$} +$$\int{e^{ax}}= +\frac{e^{ax}}{a} +$$ +<<*>>= +)spool schaum25.output +)set message test on +)set message auto off +)clear all + +--S 1 of 16 +aa:=integrate(e^(a*x),x) +--R +--R +--R a x log(e) +--R %e +--R (1) ------------ +--R a log(e) +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.510~~~~~$\displaystyle +\int{xe^{ax}}~dx$} +$$\int{xe^{ax}}= +\frac{e^{ax}}{x}\left(x-\frac{1}{a}\right) +$$ +<<*>>= +)clear all + +--S 2 of 16 +aa:=integrate(x*e^(a*x),x) +--R +--R +--R a x log(e) +--R (a x log(e) - 1)%e +--R (1) ---------------------------- +--R 2 2 +--R a log(e) +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.511~~~~~$\displaystyle +\int{x^2e^{ax}}~dx$} +$$\int{x^2e^{ax}}= +\frac{e^{ax}}{x}\left(x^2-\frac{2x}{a}+\frac{2}{a^2}\right) +$$ +<<*>>= +)clear all + +--S 3 of 16 +aa:=integrate(x^2*e^(a*x),x) +--R +--R +--R 2 2 2 a x log(e) +--R (a x log(e) - 2a x log(e) + 2)%e +--R (1) ------------------------------------------- +--R 3 3 +--R a log(e) +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.512~~~~~$\displaystyle +\int{x^ne^{ax}}~dx$} +$$\begin{array}{rl} +\displaystyle\int{x^ne^{ax}}=& +\displaystyle +\frac{x^ne^{ax}}{a}-\frac{n}{a}\int{x^{n-1}e^{ax}}\\ +\\ +&\displaystyle +=\frac{e^{ax}}{x}\left(x^n-\frac{nx^{n-1}}{a}+\frac{n(n-1)x^{n-2}}{a^2} +-\cdots \frac{(-1)^nn!}{a^n}\right) +\\ +&\hbox{\hskip 5cm}{\rm\ if\ }n={\rm positive integer} +\end{array} +$$ +<<*>>= +)clear all + +--S 4 of 16 +aa:=integrate(x^n*e^(a*x),x) +--R +--R +--R x +--I ++ n %I a +--I (1) | %I e d%I +--R ++ +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.513~~~~~$\displaystyle +\int{\frac{e^{ax}}{x}}~dx$} +$$\int{\frac{e^{ax}}{x}}= +\ln{x}+\frac{ax}{1\cdot 1!}+\frac{(ax)^2}{2\cdot 2!} ++\frac{(ax)^3}{3\cdot 3!}+\cdots +$$ +<<*>>= +)clear all + +--S 5 of 16 +aa:=integrate(e^(a*x)/x,x) +--R +--R +--R (1) Ei(a x log(e)) +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.514~~~~~$\displaystyle +\int{\frac{e^{ax}}{x^n}}~dx$} +$$\int{\frac{e^{ax}}{x^n}}= +\frac{-e^{ax}}{(n-1)x^{n-1}}+\frac{a}{n-1}\int{\frac{e^{ax}}{x^{n-1}}} +$$ +<<*>>= +)clear all + +--S 6 of 16 +aa:=integrate(e^(a*x)/x^n,x) +--R +--R +--I x %I a +--R ++ e +--I (1) | ----- d%I +--R ++ n +--I %I +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.515~~~~~$\displaystyle +\int{\frac{dx}{p+qe^{ax}}}~dx$} +$$\int{\frac{1}{p+qe^{ax}}}= +\frac{x}{p}-\frac{1}{ap}\ln(p+qe^{ax}) +$$ +<<*>>= +)clear all + +--S 7 of 16 +aa:=integrate(1/(p+q*e^(a*x)),x) +--R +--R +--R a x log(e) +--R - log(q %e + p) + a x log(e) +--R (1) -------------------------------------- +--R a p log(e) +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.516~~~~~$\displaystyle +\int{\frac{dx}{(p+qe^{ax})^2}}~dx$} +$$\int{\frac{dx}{(p+qe^{ax})^2}}= +\frac{x}{p^2}+\frac{1}{ap(p+qe^{ax})}-\frac{1}{ap^2}\ln(p+qe^{ax}) +$$ +<<*>>= +)clear all + +--S 8 of 16 +aa:=integrate(1/(p+q*e^(a*x)),x) +--R +--R +--R a x log(e) +--R - log(q %e + p) + a x log(e) +--R (1) -------------------------------------- +--R a p log(e) +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.517~~~~~$\displaystyle +\int{\frac{dx}{pe^{ax}+qe^{ax}}}~dx$} +$$\int{\frac{dx}{pe^{ax}+qe^{ax}}}= +\left\{ +\begin{array}{l} +\displaystyle +\frac{1}{a\sqrt{pq}}\tan^{-1}\left(\sqrt{\frac{p}{q}}e^{ax}\right)\\ +\\ +\displaystyle +\frac{1}{2a\sqrt{-pq}} +\ln\left(\frac{e^{ax}-\sqrt{-q/p}}{e^{ax}+\sqrt{-q/p}}\right) +\end{array} +\right. +$$ +<<*>>= +)clear all + +--S 9 of 16 +aa:=integrate(1/(p*e^(a*x)+q*e^(a*x)),x) +--R +--R +--R 1 +--R (1) - ----------------------------- +--R a x log(e) +--R (a q + a p)log(e)%e +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.518~~~~~$\displaystyle +\int{e^{ax}\sin{bx}}~dx$} +$$\int{e^{ax}\sin{bx}}= +\frac{e^{ax}(a\sin{bx}-b\cos{bx})}{a^2+b^2} +$$ +<<*>>= +)clear all + +--S 10 of 16 +aa:=integrate(e^(a*x)*sin(b*x),x) +--R +--R +--R a x log(e) +--R (a log(e)sin(b x) - b cos(b x))%e +--R (1) ------------------------------------------- +--R 2 2 2 +--R a log(e) + b +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.519~~~~~$\displaystyle +\int{e^{ax}\cos{bx}}~dx$} +$$\int{e^{ax}\cos{bx}}= +\frac{e^{ax}(a\cos{bx}-b\sin{bx})}{a^2+b^2} +$$ +<<*>>= +)clear all + +--S 11 of 16 +aa:=integrate(e^(a*x)*cos(b*x),x) +--R +--R +--R a x log(e) +--R (b sin(b x) + a cos(b x)log(e))%e +--R (1) ------------------------------------------- +--R 2 2 2 +--R a log(e) + b +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.520~~~~~$\displaystyle +\int{xe^{ax}\sin{bx}}~dx$} +$$\int{xe^{ax}\sin{bx}}= +\frac{xe^{ax}(a\sin{bx}-b\cos{bx})}{a^2+b^2} +-\frac{e^{ax}\left((a^2-b^2)\sin{bx}-2ab\cos{bx}\right)}{(a^2+b^2)^2} +$$ +<<*>>= +)clear all + +--S 12 of 16 +aa:=integrate(x*e^(a*x)*sin(b*x),x) +--R +--R +--R (1) +--R 3 3 2 2 2 2 +--R (a x log(e) - a log(e) + a b x log(e) + b )sin(b x) +--R + +--R 2 2 3 +--R - a b x cos(b x)log(e) + 2a b cos(b x)log(e) - b x cos(b x) +--R * +--R a x log(e) +--R %e +--R / +--R 4 4 2 2 2 4 +--R a log(e) + 2a b log(e) + b +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.521~~~~~$\displaystyle +\int{xe^{ax}\cos{bx}}~dx$} +$$\int{xe^{ax}\cos{bx}}= +\frac{xe^{ax}(a\cos{bx}-b\sin{bx})}{a^2+b^2} +-\frac{e^{ax}\left((a^2-b^2)\cos{bx}-2ab\sin{bx}\right)}{(a^2+b^2)^2} +$$ +<<*>>= +)clear all + +--S 13 of 16 +aa:=integrate(x*e^(a*x)*cos(b*x),x) +--R +--R +--R (1) +--R 2 2 3 3 3 +--R (a b x log(e) - 2a b log(e) + b x)sin(b x) + a x cos(b x)log(e) +--R + +--R 2 2 2 2 +--R - a cos(b x)log(e) + a b x cos(b x)log(e) + b cos(b x) +--R * +--R a x log(e) +--R %e +--R / +--R 4 4 2 2 2 4 +--R a log(e) + 2a b log(e) + b +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.522~~~~~$\displaystyle +\int{e^{ax}\ln{x}}~dx$} +$$\int{e^{ax}\ln{x}}= +\frac{e^{ax}\ln{x}}{a}-\frac{1}{a}\int{\frac{e^{ax}}{x}} +$$ +<<*>>= +)clear all + +--S 14 of 16 +aa:=integrate(e^(a*x)*log(x),x) +--R +--R +--R x +--I ++ %I a +--I (1) | log(%I)e d%I +--R ++ +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.523~~~~~$\displaystyle +\int{e^{ax}\sin^n{bx}}~dx$} +$$\int{e^{ax}\sin^n{bx}}= +\frac{e^{ax}\sin^{n-1}{bx}}{a^2+n^2b^2}(a\sin{bx}-nb\cos{bx}) ++\frac{n(n-1)b^2}{a^2+n^2b^2}\int{e^{ax}\sin^{n-2}{bx}} +$$ +<<*>>= +)clear all + +--S 15 of 16 +aa:=integrate(e^(a*x)*sin(b*x)^n,x) +--R +--R +--R x +--I ++ %I a n +--I (1) | e sin(%I b) d%I +--R ++ +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.524~~~~~$\displaystyle +\int{e^{ax}\cos^n{bx}}~dx$} +$$\int{e^{ax}\cos^n{bx}}= +\frac{e^{ax}\cos^{n-1}{bx}}{a^2+n^2b^2}(a\cos{bx}-nb\sin{bx}) ++\frac{n(n-1)b^2}{a^2+n^2b^2}\int{e^{ax}\cos^{n-2}{bx}} +$$ +<<*>>= +)clear all + +--S 16 of 16 +aa:=integrate(e^(a*x)*cos(b*x)^n,x) +--R +--R +--R x +--I ++ %I a n +--I (1) | e cos(%I b) 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 p85 +\end{thebibliography} +\end{document}
_______________________________________________ Axiom-developer mailing list Axiom-developer@nongnu.org http://lists.nongnu.org/mailman/listinfo/axiom-developer