diff --git a/changelog b/changelog index 102ba8b..13f5f2e 100644 --- a/changelog +++ b/changelog @@ -1,3 +1,10 @@ +20080403 tpd src/input/Makefile add integration regression testing +20080403 tpd src/input/schaum18.input integrals of cos(ax) +20080403 tpd src/input/schaum17.input ignore gensyms +20080403 tpd src/input/schaum16.input ignore gensyms +20080403 tpd src/input/schaum14.input ignore gensyms +20080403 tpd src/input/schaum13.input ignore gensyms +20080403 tpd src/input/schaum12.input ignore gensyms 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 diff --git a/src/input/Makefile.pamphlet b/src/input/Makefile.pamphlet index 6de47f3..dbeb3d7 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 \ + schaum17.regress schaum18.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}/schaum17.input ${OUT}/schaum18.input \ ${OUT}/saddle.input \ ${OUT}/scherk.input ${OUT}/scope.input ${OUT}/seccsc.input \ ${OUT}/segbind.input ${OUT}/seg.input ${OUT}/series2.input \ @@ -944,7 +944,7 @@ DOCFILES= \ ${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}/schaum17.input.dvi ${DOC}/schaum18.input.dvi \ ${DOC}/s01eaf.input.dvi ${DOC}/s13aaf.input.dvi \ ${DOC}/s13acf.input.dvi ${DOC}/s13adf.input.dvi \ ${DOC}/s14aaf.input.dvi ${DOC}/s14abf.input.dvi \ diff --git a/src/input/schaum12.input.pamphlet b/src/input/schaum12.input.pamphlet index 6669ade..644ebed 100644 --- a/src/input/schaum12.input.pamphlet +++ b/src/input/schaum12.input.pamphlet @@ -189,10 +189,10 @@ aa:=integrate(x^m/(a*x^2+b*x+c),x) --R --R --R x m ---R ++ %N ---R (1) | --------------- d%N +--I ++ %N +--I (1) | --------------- d%N --R ++ 2 ---R c + %N b + %N a +--I c + %N b + %N a --R Type: Union(Expression Integer,...) --E @ @@ -326,9 +326,9 @@ aa:=integrate(1/(x^n*(a*x^2+b*x+c)),x) --R --R x --R ++ 1 ---R (1) | -------------------- d%N +--I (1) | -------------------- d%N --R ++ 2 n ---R (c + %N b + %N a)%N +--I (c + %N b + %N a)%N --R Type: Union(Expression Integer,...) --E @ @@ -519,10 +519,10 @@ aa:=integrate(x^m/(a*x^2+b*x+c)^n,x) --R --R --R x m ---R ++ %N ---R (1) | ------------------ d%N +--I ++ %N +--I (1) | ------------------ d%N --R ++ 2 n ---R (c + %N b + %N a) +--I (c + %N b + %N a) --R Type: Union(Expression Integer,...) --E @ @@ -546,10 +546,10 @@ aa:=integrate(x^(2*n-1)/(a*x^2+b*x+c)^n,x) --R --R --R x 2n - 1 ---R ++ %N ---R (1) | ------------------ d%N +--I ++ %N +--I (1) | ------------------ d%N --R ++ 2 n ---R (c + %N b + %N a) +--I (c + %N b + %N a) --R Type: Union(Expression Integer,...) --E @ @@ -762,9 +762,9 @@ aa:=integrate(1/(x^m*(a*x^2+b*x+c)^n),x) --R --R x --R ++ 1 ---R (1) | --------------------- d%N +--I (1) | --------------------- d%N --R ++ m 2 n ---R %N (c + %N b + %N a) +--I %N (c + %N b + %N a) --R Type: Union(Expression Integer,...) --E diff --git a/src/input/schaum13.input.pamphlet b/src/input/schaum13.input.pamphlet index 04551e6..eb1a6eb 100644 --- a/src/input/schaum13.input.pamphlet +++ b/src/input/schaum13.input.pamphlet @@ -1167,7 +1167,7 @@ aa:=integrate((a*x^2+b*x+c)^(n+1/2),x) --R 2n + 1 --R x ------ --R ++ 2 2 ---R (1) | (c + %N b + %N a) d%N +--I (1) | (c + %N b + %N a) d%N --R ++ --R Type: Union(Expression Integer,...) --E @@ -1189,7 +1189,7 @@ aa:=integrate(x*(a*x^2+b*x+c)^(n+1/2),x) --R 2n + 1 --R x ------ --R ++ 2 2 ---R (1) | %N (c + %N b + %N a) d%N +--I (1) | %N (c + %N b + %N a) d%N --R ++ --R Type: Union(Expression Integer,...) --E @@ -1214,11 +1214,11 @@ aa:=integrate(1/(a*x^2+b*x+c)^(n+1/2),x) --R --R x --R ++ 1 ---R (1) | ----------------------- d%N +--I (1) | ----------------------- d%N --R ++ 2n + 1 --R ------ --R 2 2 ---R (c + %N b + %N a) +--I (c + %N b + %N a) --R Type: Union(Expression Integer,...) --E @ @@ -1245,11 +1245,11 @@ aa:=integrate(1/(x*(a*x^2+b*x+c)^(n+1/2)),x) --R --R x --R ++ 1 ---R (1) | -------------------------- d%N +--I (1) | -------------------------- d%N --R ++ 2n + 1 --R ------ --R 2 2 ---R %N (c + %N b + %N a) +--I %N (c + %N b + %N a) --R Type: Union(Expression Integer,...) --E diff --git a/src/input/schaum14.input.pamphlet b/src/input/schaum14.input.pamphlet index 560ab37..532a38a 100644 --- a/src/input/schaum14.input.pamphlet +++ b/src/input/schaum14.input.pamphlet @@ -264,10 +264,10 @@ aa:=integrate(x^m/(x^3+a^3),x) --R --R --R x m ---R ++ %L ---R (1) | -------- d%L +--I ++ %L +--I (1) | -------- d%L --R ++ 3 3 ---R a + %L +--I a + %L --R Type: Union(Expression Integer,...) --E @ @@ -286,9 +286,9 @@ aa:=integrate(1/(x^n*(x^3+a^3)),x) --R --R x --R ++ 1 ---R (1) | ------------- d%L +--I (1) | ------------- d%L --R ++ 3 3 n ---R (a + %L )%L +--I (a + %L )%L --R Type: Union(Expression Integer,...) --E diff --git a/src/input/schaum16.input.pamphlet b/src/input/schaum16.input.pamphlet index 2246f9a..f6b849c 100644 --- a/src/input/schaum16.input.pamphlet +++ b/src/input/schaum16.input.pamphlet @@ -64,10 +64,10 @@ aa:=integrate(x^m/(x^n+a^n)^r,x) --R --R --R x m ---R ++ %J ---R (1) | ----------- d%J +--I ++ %J +--I (1) | ----------- d%J --R ++ n n r ---R (a + %J ) +--I (a + %J ) --R Type: Union(Expression Integer,...) --E @ @@ -87,9 +87,9 @@ aa:=integrate(1/(x^m*(x^n+a^n)^r),x) --R --R x --R ++ 1 ---R (1) | -------------- d%J +--I (1) | -------------- d%J --R ++ m n n r ---R %J (a + %J ) +--I %J (a + %J ) --R Type: Union(Expression Integer,...) --E @ @@ -187,10 +187,10 @@ aa:=integrate(x^m/(x^n-a^n)^r,x) --R --R --R x m ---R ++ %J ---R (1) | ------------- d%J +--I ++ %J +--I (1) | ------------- d%J --R ++ n n r ---R (- a + %J ) +--I (- a + %J ) --R Type: Union(Expression Integer,...) --E @ @@ -210,9 +210,9 @@ aa:=integrate(1/(x^m*(x^n-a^n)^r),x) --R --R x --R ++ 1 ---R (1) | ---------------- d%J +--I (1) | ---------------- d%J --R ++ m n n r ---R %J (- a + %J ) +--I %J (- a + %J ) --R Type: Union(Expression Integer,...) --E @ @@ -274,10 +274,10 @@ aa:=integrate(x^(p-1)/(x^(2*m)+a^(2*m)),x) --R --R --R x p - 1 ---R ++ %J ---R (1) | ---------- d%J +--I ++ %J +--I (1) | ---------- d%J --R ++ 2m 2m ---R a + %J +--I a + %J --R Type: Union(Expression Integer,...) --E @ @@ -304,10 +304,10 @@ aa:=integrate(x^(p-1)/(x^(2*m)-a^(2*m)),x) --R --R --R x p - 1 ---R ++ %J ---R (1) | - ---------- d%J +--I ++ %J +--I (1) | - ---------- d%J --R ++ 2m 2m ---R a - %J +--I a - %J --R Type: Union(Expression Integer,...) --E @ @@ -338,10 +338,10 @@ 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 +--I ++ %J +--I (1) | ------------------ d%J --R ++ 2m + 1 2m + 1 ---R a + %J +--I a + %J --R Type: Union(Expression Integer,...) --E @ @@ -374,10 +374,10 @@ 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 +--I ++ %J +--I (1) | - ------------------ d%J --R ++ 2m + 1 2m + 1 ---R a - %J +--I a - %J --R Type: Union(Expression Integer,...) --E diff --git a/src/input/schaum17.input.pamphlet b/src/input/schaum17.input.pamphlet index 7ab11d7..fd40132 100644 --- a/src/input/schaum17.input.pamphlet +++ b/src/input/schaum17.input.pamphlet @@ -32,7 +32,7 @@ aa:=integrate(sin(a*x),x) \section{\cite{1}:14.340~~~~~$\displaystyle \int{x\sin{ax}~dx}$} $$\int{x\sin{ax}}= -\frac{sin{ax}}{a^2}-\frac{x\cos{ax}}{a} +\frac{\sin{ax}}{a^2}-\frac{x\cos{ax}}{a} $$ <<*>>= )clear all @@ -122,10 +122,10 @@ aa:=integrate(sin(a*x)/x^2,x) --R --R --R x ---R ++ sin(%I a) ---R (1) | --------- d%I +--I ++ sin(%I a) +--I (1) | --------- d%I --R ++ 2 ---R %I +--I %I --R Type: Union(Expression Integer,...) --E @ @@ -167,9 +167,9 @@ aa:=integrate(x/sin(a*x),x) --R --R --R x ---R ++ %I ---R (1) | --------- d%I ---R ++ sin(%I a) +--I ++ %I +--I (1) | --------- d%I +--I ++ sin(%I a) --R Type: Union(Expression Integer,...) --E @ @@ -675,7 +675,7 @@ aa:=integrate(x^m*sin(a*x),x) --R --R x --R ++ m ---R (1) | sin(%I a)%I d%I +--I (1) | sin(%I a)%I d%I --R ++ --R Type: Union(Expression Integer,...) --E @@ -694,10 +694,10 @@ aa:=integrate(sin(a*x)/x^n,x) --R --R --R x ---R ++ sin(%I a) ---R (1) | --------- d%I +--I ++ sin(%I a) +--I (1) | --------- d%I --R ++ n ---R %I +--I %I --R Type: Union(Expression Integer,...) --E @ @@ -716,7 +716,7 @@ aa:=integrate(sin(a*x)^n,x) --R --R x --R ++ n ---R (1) | sin(%I a) d%I +--I (1) | sin(%I a) d%I --R ++ --R Type: Union(Expression Integer,...) --E @@ -737,9 +737,9 @@ aa:=integrate(1/(sin(a*x))^n,x) --R --R x --R ++ 1 ---R (1) | ---------- d%I +--I (1) | ---------- d%I --R ++ n ---R sin(%I a) +--i sin(%I a) --R Type: Union(Expression Integer,...) --E @ @@ -759,10 +759,10 @@ aa:=integrate(x/sin(a*x)^n,x) --R --R --R x ---R ++ %I ---R (1) | ---------- d%I +--I ++ %I +--I (1) | ---------- d%I --R ++ n ---R sin(%I a) +--I sin(%I a) --R Type: Union(Expression Integer,...) --E diff --git a/src/input/schaum18.input.pamphlet b/src/input/schaum18.input.pamphlet new file mode 100644 index 0000000..3cb4766 --- /dev/null +++ b/src/input/schaum18.input.pamphlet @@ -0,0 +1,754 @@ +\documentclass{article} +\usepackage{axiom} +\begin{document} +\title{\$SPAD/input schaum18.input} +\author{Timothy Daly} +\maketitle +\eject +\tableofcontents +\eject +\section{\cite{1}:14.369~~~~~$\displaystyle +\int{\cos ax ~dx}$} +$$\int{\cos ax}= +\frac{\sin{ax}}{a} +$$ +<<*>>= +)spool schaum18.output +)set message test on +)set message auto off +)clear all + +--S 1 of 30 +aa:=integrate(cos(a*x),x) +--R +--R +--R sin(a x) +--R (1) -------- +--R a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.370~~~~~$\displaystyle +\int{x\cos{ax}~dx}$} +$$\int{x\cos{ax}}= +\frac{\cos{ax}}{a^2}+\frac{x\sin{ax}}{a} +$$ +<<*>>= +)clear all + +--S 2 of 30 +aa:=integrate(x*cos(a*x),x) +--R +--R +--R a x sin(a x) + cos(a x) +--R (1) ----------------------- +--R 2 +--R a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.371~~~~~$\displaystyle +\int{x^2\cos{ax}~dx}$} +$$\int{x^2\cos{ax}}= +\frac{2x}{a^2}\cos{ax}+\left(\frac{x^2}{a}-\frac{2}{a^3}\right)\sin{ax} +$$ +<<*>>= +)clear all + +--S 3 of 30 +aa:=integrate(x^2*cos(a*x),x) +--R +--R +--R 2 2 +--R (a x - 2)sin(a x) + 2a x cos(a x) +--R (1) ---------------------------------- +--R 3 +--R a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.372~~~~~$\displaystyle +\int{x^3\cos{ax}~dx}$} +$$\int{x^3\cos{ax}}= +\left(\frac{3x^2}{a^2}-\frac{6}{a^4}\right)\cos{ax} ++\left(\frac{x^3}{a}-\frac{6x}{a^3}\right)\sin{ax} +$$ +<<*>>= +)clear all + +--S 4 of 30 +aa:=integrate(x^3*cos(a*x),x) +--R +--R +--R 3 3 2 2 +--R (a x - 6a x)sin(a x) + (3a x - 6)cos(a x) +--R (1) ------------------------------------------- +--R 4 +--R a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.373~~~~~$\displaystyle +\int{\frac{\cos{ax}}{x}}~dx$} +$$\int{\frac{\cos{ax}}{x}}= +\ln{x}-\frac{(ax)^2}{2\cdot 2!}+\frac{(ax)^4}{4\cdot 4!} +-\frac{(ax)^6}{6\cdot 6!}+\cdots +$$ +<<*>>= +)clear all + +--S 5 of 30 +aa:=integrate(cos(x)/x,x) +--R +--R +--R (1) Ci(x) +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.374~~~~~$\displaystyle +\int{\frac{\cos{ax}}{x^2}}~dx$} +$$\int{\frac{\cos{ax}}{x^2}}= +-\frac{\cos{ax}}{x}-a\int{\frac{\sin{ax}}{x}} +$$ +<<*>>= +)clear all + +--S 6 of 30 +aa:=integrate(cos(a*x)/x^2,x) +--R +--R +--R x +--I ++ cos(%I a) +--I (1) | --------- d%I +--R ++ 2 +--I %I +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.375~~~~~$\displaystyle +\int{\frac{dx}{\cos{ax}}}$} +$$\int{\frac{1}{\cos{ax}}}= +\frac{1}{a}\ln(\sec{ax}-\tan{ax})= +\frac{1}{a}\ln\tan\left(\frac{\pi}{4}+\frac{ax}{2}\right) +$$ +<<*>>= +)clear all + +--S 7 of 30 +aa:=integrate(1/cos(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}:14.376~~~~~$\displaystyle +\int{\frac{x~dx}{\cos{ax}}}$} +$$\int{\frac{x}{\cos{ax}}}= +\frac{1}{a^2}\left\{ +\frac{(ax)^2}{2}+\frac{(ax)^4}{8}+\frac{5(ax)^6}{144}+\cdots+ +\frac{E_n(ax)^{2n+2}}{(2n+2)(2n)!}+\cdots +\right\} +$$ +<<*>>= +)clear all + +--S 8 of 30 +aa:=integrate(x/cos(a*x),x) +--R +--R +--R x +--I ++ %I +--I (1) | --------- d%I +--I ++ cos(%I a) +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.377~~~~~$\displaystyle +\int{\cos^2{ax}}~dx$} +$$\int{\cos^2{ax}}= +\frac{x}{2}+\frac{\sin{2ax}}{4a} +$$ +<<*>>= +)clear all + +--S 9 of 30 +aa:=integrate(cos(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}:14.378~~~~~$\displaystyle +\int{x\cos^2{ax}}~dx$} +$$\int{x\cos^2{ax}}= +\frac{x^2}{4}+\frac{x\sin{2ax}}{4a}+\frac{\cos{2ax}}{8a^2} +$$ +<<*>>= +)clear all + +--S 10 of 30 +aa:=integrate(x*cos(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}:14.379~~~~~$\displaystyle +\int{\cos^3{ax}}~dx$} +$$\int{\cos^3{ax}}= +\frac{\sin{ax}}{a}-\frac{\sin^3{ax}}{3a} +$$ +<<*>>= +)clear all + +--S 11 of 30 +aa:=integrate(cos(a*x)^3,x) +--R +--R +--R 2 +--R (cos(a x) + 2)sin(a x) +--R (1) ----------------------- +--R 3a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.380~~~~~$\displaystyle +\int{\cos^4{ax}}~dx$} +$$\int{\cos^4{ax}}= +\frac{3x}{8}+\frac{\sin{2ax}}{4a}+\frac{\sin{4ax}}{32a} +$$ +<<*>>= +)clear all + +--S 12 of 30 +aa:=integrate(cos(a*x)^4,x) +--R +--R +--R 3 +--R (2cos(a x) + 3cos(a x))sin(a x) + 3a x +--R (1) --------------------------------------- +--R 8a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.381~~~~~$\displaystyle +\int{\frac{dx}{\cos^2{ax}}}$} +$$\int{\frac{1}{\cos^2{ax}}}= +\frac{\tan{ax}}{a} +$$ +<<*>>= +)clear all + +--S 13 of 30 +aa:=integrate(1/cos(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}:14.382~~~~~$\displaystyle +\int{\frac{dx}{\cos^3{ax}}}$} +$$\int{\frac{1}{\cos^3{ax}}}= +\frac{\sin{ax}}{2a\cos^2{ax}} ++\frac{1}{2a}\ln\tan\left(\frac{\pi}{4}+\frac{ax}{2}\right) +$$ +<<*>>= +)clear all + +--S 14 of 30 +aa:=integrate(1/cos(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}:14.383~~~~~$\displaystyle +\int{\cos{px}\cos{qx}}~dx$} +$$\int{\cos{ax}\cos{px}}= +\frac{\sin{(a-p)x}}{2(a-p)}+\frac{\sin{(a+p)x}}{2(a+p)} +$$ +<<*>>= +)clear all + +--S 15 of 30 +aa:=integrate(cos(p*x)*cos(q*x),x) +--R +--R +--R q cos(p x)sin(q x) - p cos(q x)sin(p x) +--R (1) --------------------------------------- +--R 2 2 +--R q - p +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.384~~~~~$\displaystyle +\int{\frac{dx}{1-\cos{ax}}}$} +$$\int{\frac{1}{1-\cos{ax}}}= +-\frac{1}{a}\cot\frac{ax}{2} +$$ +<<*>>= +)clear all + +--S 16 of 30 +aa:=integrate(1/(1-cos(a*x)),x) +--R +--R +--R - cos(a x) - 1 +--R (1) -------------- +--R a sin(a x) +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.385~~~~~$\displaystyle +\int{\frac{x~dx}{1-\cos{ax}}}$} +$$\int{\frac{x}{1-\cos{ax}}}= +-\frac{x}{a}\cot\frac{ax}{2} ++\frac{2}{a^2}\ln~\sin\frac{ax}{2} +$$ +<<*>>= +)clear all + +--S 17 of 30 +aa:=integrate(x/(1-cos(ax)),x) +--R +--R +--R 2 +--R x +--R (1) - ------------ +--R 2cos(ax) - 2 +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.386~~~~~$\displaystyle +\int{\frac{dx}{1+\cos{ax}}}$} +$$\int{\frac{1}{1+\cos{ax}}}= +\frac{1}{a}\tan\frac{ax}{2} +$$ +<<*>>= +)clear all + +--S 18 of 30 +aa:=integrate(1/(1+cos(ax)),x) +--R +--R +--R x +--R (1) ----------- +--R cos(ax) + 1 +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.387~~~~~$\displaystyle +\int{\frac{x~dx}{1+\cos{ax}}}$} +$$\int{\frac{x}{1+\cos{ax}}}= +\frac{x}{a}\tan\frac{ax}{2} ++\frac{2}{a^2}\ln~\cos\frac{ax}{2} +$$ +<<*>>= +)clear all + +--S 19 of 30 +aa:=integrate(x/(1+cos(a*x)),x) +--R +--R +--R 2 +--R (- cos(a x) - 1)log(------------) + a x sin(a x) +--R cos(a x) + 1 +--R (1) ------------------------------------------------ +--R 2 2 +--R a cos(a x) + a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.388~~~~~$\displaystyle +\int{\frac{dx}{(1-\cos{ax})^2}}$} +$$\int{\frac{1}{(1-\cos{ax})^2}}= +-\frac{1}{2a}\cot\frac{ax}{2} +-\frac{1}{6a}\cot^3\frac{ax}{2} +$$ +<<*>>= +)clear all + +--S 20 of 30 +aa:=integrate(1/(1-cos(a*x))^2,x) +--R +--R +--R 2 +--R - cos(a x) + cos(a x) + 2 +--R (1) -------------------------- +--R (3a cos(a x) - 3a)sin(a x) +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.389~~~~~$\displaystyle +\int{\frac{dx}{(1+\cos{ax})^2}}$} +$$\int{\frac{1}{(1+\cos{ax})^2}}= +\frac{1}{2a}\tan\frac{ax}{2} ++\frac{1}{6a}\tan^3\frac{ax}{2} +$$ +<<*>>= +)clear all + +--S 21 of 30 +aa:=integrate(1/(1+cos(a*x))^2,x) +--R +--R +--R (cos(a x) + 2)sin(a x) +--R (1) ------------------------------- +--R 2 +--R 3a cos(a x) + 6a cos(a x) + 3a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.390~~~~~$\displaystyle +\int{\frac{dx}{p+q\cos{ax}}}$} +$$\int{\frac{1}{p+q\cos{ax}}}= +\left\{ +\begin{array}{l} +\displaystyle +\frac{2}{a\sqrt{p^2-q^q}} +\tan^{-1}\sqrt{(p-q)/(p+q)}\tan\frac{1}{2}ax +\\ +\displaystyle +\frac{1}{a\sqrt{q^2-p^2}}\ln\left( +\frac{\tan\frac{1}{2}ax+\sqrt{(q+p)/(q-p)}} +{\tan\frac{1}{2}ax-\sqrt{(q+p)(q-p)}}\right) +\end{array} +\right. +$$ +<<*>>= +)clear all + +--S 22 of 30 +aa:=integrate(1/(p+q*cos(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) +--R log(--------------------------------------------------) +--R q cos(a x) + p +--R [-------------------------------------------------------, +--R +-------+ +--R | 2 2 +--R a\|q - p +--R +---------+ +--R | 2 2 +--R sin(a x)\|- q + p +--R 2atan(-----------------------) +--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}:14.391~~~~~$\displaystyle +\int{\frac{dx}{(p+q\cos{ax})^2}}$} +$$\int{\frac{1}{(p+q\cos{ax})^2}}= +\frac{q\sin{ax}}{a(q^2-p^2)(p+q\cos{ax})} +-\frac{p}{q^2-p^2}\int{\frac{1}{p+q\cos{ax}}} +$$ +<<*>>= +)clear all + +--S 23 of 30 +aa:=integrate(1/(p+q*cos(a*x))^2,x) +--R +--R +--R (1) +--R [ +--R 2 +--R (p q cos(a x) + p ) +--R * +--R +-------+ +--R | 2 2 2 2 +--R (- p cos(a x) - q)\|q - p + (q - p )sin(a x) +--R log(------------------------------------------------) +--R q cos(a x) + p +--R + +--R +-------+ +--R | 2 2 +--R q sin(a x)\|q - p +--R / +--R +-------+ +--R 3 2 2 3 | 2 2 +--R ((a q - a p q)cos(a x) + a p q - a p )\|q - p +--R , +--R +--R +---------+ +--R | 2 2 +--R 2 sin(a x)\|- q + p +--R (- 2p q cos(a x) - 2p )atan(-----------------------) +--R (q + p)cos(a x) + q + p +--R + +--R +---------+ +--R | 2 2 +--R q sin(a x)\|- q + p +--R / +--R +---------+ +--R 3 2 2 3 | 2 2 +--R ((a q - a p q)cos(a x) + a p q - a p )\|- q + p +--R ] +--R Type: Union(List Expression Integer,...) +--E +@ + +\section{\cite{1}:14.392~~~~~$\displaystyle +\int{\frac{dx}{p^2+q^2\cos^2{ax}}}$} +$$\int{\frac{1}{p^2+q^2\cos^2{ax}}}= +\frac{1}{ap\sqrt{p^2+q^2}}\tan^{-1}\frac{p\tan{ax}}{\sqrt{p^2+q^2}} +$$ +<<*>>= +)clear all + +--S 24 of 30 +aa:=integrate(1/(p^2+a^2*cos(a*x)),x) +--R +--R +--R (1) +--R +---------+ +--R 2 2 | 4 4 4 4 +--R (- p cos(a x) - a )\|- p + a + (p - a )sin(a x) +--R log(---------------------------------------------------) +--R 2 2 +--R a cos(a x) + p +--R [--------------------------------------------------------, +--R +---------+ +--R | 4 4 +--R a\|- p + a +--R +-------+ +--R | 4 4 +--R sin(a x)\|p - a +--R 2atan(---------------------------) +--R 2 2 2 2 +--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}:14.393~~~~~$\displaystyle +\int{\frac{dx}{p^2-q^2\cos^2{ax}}}$} +$$\int{\frac{1}{p^2-q^2\cos^2{ax}}}= +\left\{ +\begin{array}{l} +\displaystyle +\frac{1}{ap\sqrt{p^2-q^2}}\tan^{-1}\frac{p\tan{ax}}{\sqrt{p^2-q^2}}\\ +\\ +\displaystyle +\frac{1}{2ap\sqrt{q^2-p^2}}\ln\left(\frac{p\tan{ax}-\sqrt{q^2-p^2}} +{p\tan{ax}+\sqrt{q^2-p^2}}\right) +\end{array} +\right. +$$ +<<*>>= +)clear all + +--S 25 of 30 +aa:=integrate(1/(p^2-q^2*cos(a*x)^2),x) +--R +--R +--R (1) +--R +-------+ +--R 2 2 2 2 | 2 2 2 3 +--R ((q - 2p )cos(a x) + p )\|q - p + (- 2p q + 2p )cos(a x)sin(a x) +--R log(----------------------------------------------------------------------) +--R 2 2 2 +--R q cos(a x) - p +--R [---------------------------------------------------------------------------, +--R +-------+ +--R | 2 2 +--R 2a p\|q - p +--R +--R +---------+ +--R | 2 2 +--R sin(a x)\|- q + p +--R atan(--------------------) +--R 2p cos(a x) + 2p +--R + +--R 2 2 2 +--R ((q + p )cos(a x) + 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}:14.394~~~~~$\displaystyle +\int{x^m\cos{ax}}~dx$} +$$\int{x^m\cos{ax}}= +\frac{x^m\sin{ax}}{a}+\frac{mx^{m-1}}{a^2}\cos{ax} +-\frac{m(m-1)}{a^2}\int{x^{m-2}\cos{ax}} +$$ +<<*>>= +)clear all + +--S 26 of 30 +aa:=integrate(x^m*cos(a*x),x) +--R +--R +--R x +--R ++ m +--I (1) | cos(%I a)%I d%I +--R ++ +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.395~~~~~$\displaystyle +\int{\frac{\cos{ax}}{x^n}}~dx$} +$$\int{\frac{\cos{ax}}{x^n}}= +-\frac{\cos{ax}}{(n-1)x^{n-1}}-\frac{a}{n-1}\int{\frac{\sin{ax}}{x^{n-1}}} +$$ +<<*>>= +)clear all + +--S 26 of 30 +aa:=integrate(cos(a*x)/x^n,x) +--R +--R +--R x +--I ++ cos(%I a) +--I (1) | --------- d%I +--R ++ n +--I %I +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.396~~~~~$\displaystyle +\int{\cos^n{ax}}~dx$} +$$\int{\cos^n{ax}}= +\frac{\sin{ax}\cos^{n-1}{ax}}{an}+\frac{n-1}{n}\int{\cos^{n-2}{ax}} +$$ +<<*>>= +)clear all + +--S 28 of 30 +aa:=integrate(cos(a*x)^n,x) +--R +--R +--R x +--R ++ n +--I (1) | cos(%I a) d%I +--R ++ +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.397~~~~~$\displaystyle +\int{\frac{1}{\cos^n{ax}}}~dx$} +$$\int{\frac{1}{\cos^n{ax}}}= +\frac{\sin{ax}}{a(n-1)\cos^{n-1}{ax}} ++\frac{n-2}{n-1}\int{\frac{1}{\cos^{n-2}{ax}}} +$$ +<<*>>= +)clear all + +--S 29 of 30 +aa:=integrate(1/(cos(a*x))^n,x) +--R +--R +--R x +--R ++ 1 +--I (1) | ---------- d%I +--R ++ n +--I cos(%I a) +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.398~~~~~$\displaystyle +\int{\frac{x~dx}{cos^n{ax}}}$} +$$\int{\frac{x}{cos^n{ax}}}= +\frac{x\sin{ax}}{a(n-1)\cos^{n-1}{ax}} +-\frac{1}{a^2(n-1)(n-2)\cos^{n-2}{ax}} ++\frac{n-2}{n-1}\int{\frac{x}{\cos^{n-2}{ax}}} +$$ +<<*>>= +)clear all + +--S 30 of 30 +aa:=integrate(x/cos(a*x)^n,x) +--R +--R +--R x +--I ++ %I +--I (1) | ---------- d%I +--R ++ n +--I cos(%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 pp77-78 +\end{thebibliography} +\end{document}