[Top][All Lists]
[Date Prev][Date Next][Thread Prev][Thread Next][Date Index][Thread Index]
[Axiom-developer] 20080406.01.tpd.patch (CATS integraion regression test
|
From: |
daly |
|
Subject: |
[Axiom-developer] 20080406.01.tpd.patch (CATS integraion regression testing) |
|
Date: |
Sun, 6 Apr 2008 12:29:18 -0500 |
More integrals
=======================================================================
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}
| [Prev in Thread] |
Current Thread |
[Next in Thread] |
- [Axiom-developer] 20080406.01.tpd.patch (CATS integraion regression testing),
daly <=