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[Axiom-developer] 20070913.01.tpd.patch
|
From: |
daly |
|
Subject: |
[Axiom-developer] 20070913.01.tpd.patch |
|
Date: |
Thu, 13 Sep 2007 22:35:25 -0500 |
This patch tests the equations 14.39-14.80 from Spiegel's Mathematical
Handbook of the Schaum's Outline Series, the 1968 edition. Each equation
is shown to give a zero difference from the book answer where possible.
Note that the book results for 14.73, 14.77, and 14.79 are incorrect.
The correct results are given here.
=========================================================================
diff --git a/changelog b/changelog
index 6ff3473..1a831be 100644
--- a/changelog
+++ b/changelog
@@ -1,3 +1,5 @@
+20070913 tpd src/input/Makefile schaum1.input added
+20070913 tpd src/input/schaum1.input added
20070909 tpd src/algebra/newton.spad included in fffg.spad
20070909 tpd src/algebra/Makefile remove newton.spad (duplicate)
20070907 tpd src/algebra/acplot.spad fix PlaneAlgebraicCurvePlot.help NOISE
diff --git a/src/input/Makefile.pamphlet b/src/input/Makefile.pamphlet
index 2123928..16a3b98 100644
--- a/src/input/Makefile.pamphlet
+++ b/src/input/Makefile.pamphlet
@@ -345,6 +345,7 @@ REGRES= algaggr.regress algbrbf.regress algfacob.regress
alist.regress \
r21bugsbig.regress r21bugs.regress radff.regress radix.regress \
realclos.regress reclos.regress repa6.regress robidoux.regress \
roman.regress roots.regress ruleset.regress rules.regress \
+ schaum1.regress \
scherk.regress scope.regress segbind.regress seg.regress \
series2.regress series.regress sersolve.regress set.regress \
sincosex.regress sint.regress skew.regress slowint.regress \
@@ -601,7 +602,8 @@ FILES= ${OUT}/algaggr.input ${OUT}/algbrbf.input
${OUT}/algfacob.input \
${OUT}/radff.input ${OUT}/radix.input ${OUT}/realclos.input \
${OUT}/reclos.input ${OUT}/regset.input \
${OUT}/robidoux.input ${OUT}/roman.input ${OUT}/roots.input \
- ${OUT}/ruleset.input ${OUT}/rules.input ${OUT}/saddle.input \
+ ${OUT}/ruleset.input ${OUT}/rules.input ${OUT}/schaum1.input \
+ ${OUT}/saddle.input \
${OUT}/scherk.input ${OUT}/scope.input \
${OUT}/segbind.input ${OUT}/seg.input ${OUT}/series2.input \
${OUT}/series.input ${OUT}/sersolve.input ${OUT}/set.input \
@@ -879,6 +881,7 @@ DOCFILES= \
${DOC}/robidoux.input.dvi ${DOC}/roman.input.dvi \
${DOC}/romnum.as.dvi ${DOC}/roots.input.dvi \
${DOC}/ruleset.input.dvi ${DOC}/rules.input.dvi \
+ ${DOC}/schaum1.input.dvi \
${DOC}/s01eaf.input.dvi ${DOC}/s13aaf.input.dvi \
${DOC}/s13acf.input.dvi ${DOC}/s13adf.input.dvi \
${DOC}/s14aaf.input.dvi ${DOC}/s14abf.input.dvi \
diff --git a/src/input/schaum1.input.pamphlet b/src/input/schaum1.input.pamphlet
new file mode 100644
index 0000000..8507428
--- /dev/null
+++ b/src/input/schaum1.input.pamphlet
@@ -0,0 +1,1265 @@
+\documentclass{article}
+\usepackage{axiom}
+\begin{document}
+\title{\$SPAD/input schaum1.input}
+\author{Timothy Daly}
+\maketitle
+\eject
+\tableofcontents
+\eject
+\section{\cite{1}:14.59~~~~~$\displaystyle\int{\frac{dx}{ax+b}~dx}$}
+$$\int{\frac{dx}{ax+b}~dx}==\frac{1}{a}~\ln(ax+b)$$
+<<*>>=
+)spool schaum1.output
+)set message test on
+)set message auto off
+)clear all
+
+--S 1
+integrate(1/(a*x+b),x)
+--R
+--R log(a x + b)
+--R (1) ------------
+--R a
+--R Type: Union(Expression
Integer,...)
+--E 1
+@
+\section{\cite{1}:14.60~~~~~$\displaystyle\int{\frac{x~dx}{ax+b}}$}
+$$\int{\frac{x~dx}{ax+b}}=\frac{x}{a}-\frac{b}{a^2}~\ln(ax+b)$$
+<<*>>=
+)clear all
+
+--S 2
+integrate(x/(a*x+b),x)
+--R
+--R
+--R - b log(a x + b) + a x
+--R (1) ----------------------
+--R 2
+--R a
+--R Type: Union(Expression
Integer,...)
+--E 2
+@
+\section{\cite{1}:14.61~~~~~$\displaystyle\int{\frac{x^2~dx}{ax+b}}$}
+$$\int{\frac{x^2~dx}{ax+b}}=
+\frac{(ax+b)^2}{2a^3}-\frac{2b(ax+b)}{a^3}+\frac{b^2}{a^3}~\ln(ax+b)$$
+<<*>>=
+)clear all
+
+--S 3
+nn:=integrate(x^2/(a*x+b),x)
+--R
+--R 2 2 2
+--R 2b log(a x + b) + a x - 2a b x
+--R (1) -------------------------------
+--R 3
+--R 2a
+--R Type: Union(Expression
Integer,...)
+--E 3
+@
+To see that these are the same answers we put the prior result over
+a common fraction:
+<<*>>=
+--S 4
+mm:=((a*x+b)^2-2*2*b*(a*x+b)+2*b^2*log(a*x+b))/(2*a^3)
+--R
+--R 2 2 2 2
+--R 2b log(a x + b) + a x - 2a b x - 3b
+--R (2) -------------------------------------
+--R 3
+--R 2a
+--R Type: Expression
Integer
+--E 4
+@
+and we take their difference:
+<<*>>=
+--S 5
+pp:=mm-nn
+--R
+--R 2
+--R 3b
+--R (3) - ---
+--R 3
+--R 2a
+--R Type: Expression
Integer
+--E 5
+@
+which is a constant with respect to x, and thus the constant C.
+<<*>>=
+--S 6
+D(pp,x)
+--R
+--R (4) 0
+--R Type: Expression
Integer
+--E 6
+@
+Alternatively we can differentiate the answers with respect to x:
+<<*>>=
+--S 7
+D(nn,x)
+--R
+--R 2
+--R x
+--R (5) -------
+--R a x + b
+--R Type: Expression
Integer
+--E 7
+@
+<<*>>=
+--S 8
+D(mm,x)
+--R
+--R 2
+--R x
+--R (6) -------
+--R a x + b
+--R Type: Expression
Integer
+--E 8
+@
+and see that they are indeed the same.
+
+\section{\cite{1}:14.62~~~~~$\displaystyle\int{\frac{x^3~dx}{ax+b}}$}
+$$\int{\frac{x^3~dx}{ax+b}}=
+\frac{(ax+b)^3}{3a^4}-\frac{3b(ax+b)^2}{2a^4}+
+\frac{3b^2(ax+b)}{a^4}-\frac{b^3}{a^4}~\ln(ax+b)$$
+<<*>>=
+)clear all
+
+--S 9
+aa:=integrate(x^3/(a*x+b),x)
+--R
+--R 3 3 3 2 2 2
+--R - 6b log(a x + b) + 2a x - 3a b x + 6a b x
+--R (1) --------------------------------------------
+--R 4
+--R 6a
+--R Type: Union(Expression
Integer,...)
+--E 9
+@
+and the book expression is:
+<<*>>=
+--S 10
+bb:=(a*x+b)^3/(3*a^4)-(3*b*(a*x+b)^2)/(2*a^4)+(3*b^2*(a*x+b))/a^4-(b^3/a^4)*log(a*x+b)
+--R
+--R 3 3 3 2 2 2 3
+--R - 6b log(a x + b) + 2a x - 3a b x + 6a b x + 11b
+--R (2) ---------------------------------------------------
+--R 4
+--R 6a
+--R Type: Expression
Integer
+--E 10
+@
+
+The difference is a constant with respect to x:
+<<*>>=
+--S 11
+aa-bb
+--R
+--R 3
+--R 11b
+--R (3) - ----
+--R 4
+--R 6a
+--R Type: Expression
Integer
+--E 11
+@
+
+If we differentiate each expression we see
+<<*>>=
+--S 12
+cc:=D(aa,x)
+--R
+--R 3
+--R x
+--R (4) -------
+--R a x + b
+--R Type: Expression
Integer
+--E 12
+@
+<<*>>=
+--S 13
+dd:=D(bb,x)
+--R
+--R 3
+--R x
+--R (5) -------
+--R a x + b
+--R Type: Expression
Integer
+--E 13
+@
+<<*>>=
+--S 14
+cc-dd
+--R
+--R (6) 0
+--R Type: Expression
Integer
+--E 14
+@
+
+\section{\cite{1}:14.63~~~~~$\displaystyle\int{\frac{dx}{x~(ax+b)}}$}
+$$\int{\frac{dx}{x~(ax+b)}}=\frac{1}{b}~\ln\left(\frac{x}{ax+b}\right)$$
+<<*>>=
+)clear all
+
+--S 15
+ff:=integrate(1/(x*(a*x+b)),x)
+--R
+--R - log(a x + b) + log(x)
+--R (1) -----------------------
+--R b
+--R Type: Union(Expression
Integer,...)
+--E 15
+@
+but we know that $$\log(a)-\log(b)=\log(\frac{a}{b})$$
+
+We can express this fact as a rule:
+<<*>>=
+--S 16
+logdiv:=rule(log(a)-log(b) == log(a/b))
+--R
+--R a
+--I (2) - log(b) + log(a) + %I == log(-) + %I
+--R b
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E 16
+@
+and use this rule to rewrite the logs into divisions:
+<<*>>=
+--S 17
+logdiv ff
+--R
+--R x
+--R log(-------)
+--R a x + b
+--R (3) ------------
+--R b
+--R Type: Expression
Integer
+--E 17
+@
+so we can see the equivalence directly.
+
+\section{\cite{1}:14.64~~~~~$\displaystyle\int{\frac{dx}{x^2~(ax+b)}}$}
+$$\int{\frac{dx}{x^2~(ax+b)}}=
+-\frac{1}{bx}+\frac{a}{b^2}~\ln\left(\frac{ax+b}{x}\right)$$
+<<*>>=
+)clear all
+
+--S 18
+aa:=integrate(1/(x^2*(a*x+b)),x)
+--R
+--R a x log(a x + b) - a x log(x) - b
+--R (1) ---------------------------------
+--R 2
+--R b x
+--R Type: Union(Expression
Integer,...)
+--E 18
+@
+
+The original form given in the book expands to:
+<<*>>=
+--S 19
+bb:=-1/(b*x)+a/b^2*log((a*x+b)/x)
+--R
+--R a x + b
+--R a x log(-------) - b
+--R x
+--R (2) --------------------
+--R 2
+--R b x
+--R Type: Expression
Integer
+--E 19
+@
+
+We can define the following rule to expand log forms:
+<<*>>=
+--S 20
+divlog:=rule(log(a/b) == log(a) - log(b))
+--R
+--R a
+--R (3) log(-) == - log(b) + log(a)
+--R b
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E 20
+@
+and apply it to the book form:
+<<*>>=
+--S 21
+cc:= divlog bb
+--R
+--R a x log(a x + b) - a x log(x) - b
+--R (4) ---------------------------------
+--R 2
+--R b x
+--R Type: Expression
Integer
+--E 21
+@
+and we can now see that the results are identical.
+<<*>>=
+--S 22
+aa-cc
+--R
+--R (5) 0
+--R Type: Expression
Integer
+--E 22
+@
+
+\section{\cite{1}:14.65~~~~~$\displaystyle\int{\frac{dx}{x^3~(ax+b)}}$}
+$$\int{\frac{dx}{x^3~(ax+b)}}=
+\frac{2ax-b}{2b^2x^2}+\frac{a^2}{b^3}~\ln\left(\frac{x}{ax+b}\right)$$
+<<*>>=
+)clear all
+--S 23
+aa:=integrate(1/(x^3*(a*x+b)),x)
+--R
+--R 2 2 2 2 2
+--R - 2a x log(a x + b) + 2a x log(x) + 2a b x - b
+--R (1) -----------------------------------------------
+--R 3 2
+--R 2b x
+--R Type: Union(Expression
Integer,...)
+--E 23
+@
+
+<<*>>=
+--S 24
+bb:=(2*a*x-b)/(2*b^2*x^2)+a^2/b^3*log(x/(a*x+b))
+--R
+--R 2 2 x 2
+--R 2a x log(-------) + 2a b x - b
+--R a x + b
+--R (2) -------------------------------
+--R 3 2
+--R 2b x
+--R Type: Expression
Integer
+--E 24
+@
+
+<<*>>=
+--S 25
+divlog:=rule(log(a/b) == log(a) - log(b))
+--R
+--R a
+--R (3) log(-) == - log(b) + log(a)
+--R b
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E 25
+@
+
+<<*>>=
+--S 26
+cc:=divlog bb
+--R
+--R 2 2 2 2 2
+--R - 2a x log(a x + b) + 2a x log(x) + 2a b x - b
+--R (4) -----------------------------------------------
+--R 3 2
+--R 2b x
+--R Type: Expression
Integer
+--E 26
+@
+
+<<*>>=
+--S 27
+cc-aa
+--R
+--R (5) 0
+--R Type: Expression
Integer
+--E 27
+@
+
+\section{\cite{1}:14.66~~~~~$\displaystyle\int{\frac{dx}{(ax+b)^2}}$}
+$$\int{\frac{dx}{(ax+b)^2}}=\frac{-1}{a~(ax+b)}$$
+<<*>>=
+)clear all
+
+--S 28
+integrate(1/(a*x+b)^2,x)
+--R
+--R 1
+--R (1) - ---------
+--R 2
+--R a x + a b
+--R Type: Union(Expression
Integer,...)
+--E 28
+@
+
+\section{\cite{1}:14.67~~~~~$\displaystyle\int{\frac{x~dx}{(ax+b)^2}}$}
+$$\int{\frac{x~dx}{(ax+b)^2}}=
+\frac{b}{a^2~(ax+b)}+\frac{1}{a^2}~\ln(ax+b)$$
+<<*>>=
+)clear all
+
+--S 29
+integrate(x/(a*x+b)^2,x)
+--R
+--R (a x + b)log(a x + b) + b
+--R (1) -------------------------
+--R 3 2
+--R a x + a b
+--R Type: Union(Expression
Integer,...)
+--E 29
+@
+and the book form expands to:
+<<*>>=
+--S 30
+b/(a^2*(a*x+b))+(1/a^2)*log(a*x+b)
+--R
+--R (a x + b)log(a x + b) + b
+--R (2) -------------------------
+--R 3 2
+--R a x + a b
+--R Type: Expression
Integer
+--E 30
+@
+
+\section{\cite{1}:14.68~~~~~$\displaystyle\int{\frac{x^2~dx}{(ax+b)^2}}$}
+$$\int{\frac{x^2~dx}{(ax+b)^2}}=
+\frac{ax+b}{a^3}-\frac{b^2}{a^3~(ax+b)}
+-\frac{2b}{a^3}~\ln(ax+b)$$
+<<*>>=
+)clear all
+
+--S 31
+aa:=integrate(x^2/(a*x+b)^2,x)
+--R
+--R 2 2 2 2
+--R (- 2a b x - 2b )log(a x + b) + a x + a b x - b
+--R (1) ------------------------------------------------
+--R 4 3
+--R a x + a b
+--R Type: Union(Expression
Integer,...)
+--E 31
+@
+and the book expression expands into
+<<*>>=
+--S 32
+bb:=(a*x+b)/a^3-b^2/(a^3*(a*x+b))-((2*b)/a^3)*log(a*x+b)
+--R
+--R 2 2 2
+--R (- 2a b x - 2b )log(a x + b) + a x + 2a b x
+--R (2) --------------------------------------------
+--R 4 3
+--R a x + a b
+--R Type: Expression
Integer
+--E 32
+@
+
+These two expressions differ by the constant
+<<*>>=
+--S 33
+aa-bb
+--R
+--R b
+--R (3) - --
+--R 3
+--R a
+--R Type: Expression
Integer
+--E 33
+@
+
+These are the same integrands as can be shown by differentiation:
+<<*>>=
+--S 34
+D(aa,x)
+--R
+--R 2
+--R x
+--R (4) ------------------
+--R 2 2 2
+--R a x + 2a b x + b
+--R Type: Expression
Integer
+--E 34
+@
+
+<<*>>=
+--S 35
+D(bb,x)
+--R
+--R 2
+--R x
+--R (5) ------------------
+--R 2 2 2
+--R a x + 2a b x + b
+--R Type: Expression
Integer
+--E 35
+@
+
+\section{\cite{1}:14.69~~~~~$\displaystyle\int{\frac{x^3~dx}{(ax+b)^2}}$}
+$$\int{\frac{x^3~dx}{(ax+b)^2}}=
+\frac{(ax+b)^2}{2a^4}-\frac{3b(ax+b)}{a^4}+\frac{b^3}{a^4(ax+b)}
++\frac{3b^2}{a^4}~\ln(ax+b)$$
+<<*>>=
+)clear all
+
+--S 36
+aa:=integrate(x^3/(a*x+b)^2,x)
+--R
+--R 2 3 3 3 2 2 2 3
+--R (6a b x + 6b )log(a x + b) + a x - 3a b x - 4a b x + 2b
+--R (1) ----------------------------------------------------------
+--R 5 4
+--R 2a x + 2a b
+--R Type: Union(Expression
Integer,...)
+--E 36
+@
+
+<<*>>=
+--S 37
+bb:=(a*x+b)^2/(2*a^4)-(3*b*(a*x+b))/a^4+b^3/(a^4*(a*x+b))+(3*b^2/a^4)*log(a*x+b)
+--R
+--R 2 3 3 3 2 2 2 3
+--R (6a b x + 6b )log(a x + b) + a x - 3a b x - 9a b x - 3b
+--R (2) ----------------------------------------------------------
+--R 5 4
+--R 2a x + 2a b
+--R Type: Expression
Integer
+--E 37
+@
+
+<<*>>=
+--S 38
+aa-bb
+--R
+--R 2
+--R 5b
+--R (3) ---
+--R 4
+--R 2a
+--R Type: Expression
Integer
+--E 38
+@
+
+<<*>>=
+--S 39
+cc:=D(aa,x)
+--R
+--R 3
+--R x
+--R (4) ------------------
+--R 2 2 2
+--R a x + 2a b x + b
+--R Type: Expression
Integer
+--E 39
+@
+
+<<*>>=
+--S 40
+dd:=D(bb,x)
+--R
+--R 3
+--R x
+--R (5) ------------------
+--R 2 2 2
+--R a x + 2a b x + b
+--R Type: Expression
Integer
+--E 40
+@
+
+<<*>>=
+--S 41
+cc-dd
+--R
+--R (6) 0
+--R Type: Expression
Integer
+--E 41
+@
+
+\section{\cite{1}:14.70~~~~~$\displaystyle\int{\frac{dx}{x~(ax+b)^2}}$}
+$$\int{\frac{dx}{x~(ax+b)^2}}=
+\frac{1}{b~(ax+b)}+\frac{1}{b^2}~\ln\left(\frac{x}{ax+b}\right)$$
+<<*>>=
+)clear all
+
+--S 42
+aa:=integrate(1/(x*(a*x+b)^2),x)
+--R
+--R (- a x - b)log(a x + b) + (a x + b)log(x) + b
+--R (1) ---------------------------------------------
+--R 2 3
+--R a b x + b
+--R Type: Union(Expression
Integer,...)
+--E 42
+@
+and the book says:
+<<*>>=
+--S 43
+bb:=(1/(b*(a*x+b))+(1/b^2)*log(x/(a*x+b)))
+--R
+--R x
+--R (a x + b)log(-------) + b
+--R a x + b
+--R (2) -------------------------
+--R 2 3
+--R a b x + b
+--R Type: Expression
Integer
+--E 43
+@
+
+So we look at the divlog rule again:
+<<*>>=
+--S 44
+divlog:=rule(log(a/b) == log(a) - log(b))
+--R
+--R a
+--R (3) log(-) == - log(b) + log(a)
+--R b
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E 44
+@
+
+we apply it:
+<<*>>=
+--S 45
+cc:=divlog bb
+--R
+--R (- a x - b)log(a x + b) + (a x + b)log(x) + b
+--R (4) ---------------------------------------------
+--R 2 3
+--R a b x + b
+--R Type: Expression
Integer
+--E 45
+@
+and we difference the two to find they are identical:
+<<*>>=
+--S 46
+cc-aa
+--R
+--R (5) 0
+--R Type: Expression
Integer
+--E 46
+@
+
+\section{\cite{1}:14.71~~~~~$\displaystyle\int{\frac{dx}{x^2~(ax+b)^2}}$}
+$$\int{\frac{dx}{x^2~(ax+b)^2}}=
+\frac{-a}{b^2~(ax+b)}-\frac{1}{b^2~x}+
+\frac{2a}{b^3}~\ln\left(\frac{ax+b}{x}\right)$$
+<<*>>=
+)clear all
+
+--S 47
+aa:=integrate(1/(x^2*(a*x+b)^2),x)
+--R
+--R 2 2 2 2
2
+--R (2a x + 2a b x)log(a x + b) + (- 2a x - 2a b x)log(x) - 2a b x - b
+--R (1)
---------------------------------------------------------------------
+--R 3 2 4
+--R a b x + b x
+--R Type: Union(Expression
Integer,...)
+--E 47
+@
+and the book says:
+<<*>>=
+--S 48
+bb:=(-a/(b^2*(a*x+b)))-(1/(b^2*x))+((2*a)/b^3)*log((a*x+b)/x)
+--R
+--R 2 2 a x + b 2
+--R (2a x + 2a b x)log(-------) - 2a b x - b
+--R x
+--R (2) ------------------------------------------
+--R 3 2 4
+--R a b x + b x
+--R Type: Expression
Integer
+--E 48
+@
+which calls for our divlog rule:
+<<*>>=
+--S 49
+divlog:=rule(log(a/b) == log(a) - log(b))
+--R
+--R a
+--R (3) log(-) == - log(b) + log(a)
+--R b
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E 49
+@
+which we use to transform the result:
+<<*>>=
+--S 50
+cc:=divlog bb
+--R
+--R 2 2 2 2
2
+--R (2a x + 2a b x)log(a x + b) + (- 2a x - 2a b x)log(x) - 2a b x - b
+--R (4)
---------------------------------------------------------------------
+--R 3 2 4
+--R a b x + b x
+--R Type: Expression
Integer
+--E 50
+@
+and we show they are identical:
+<<*>>=
+--S 51
+dd:=aa-cc
+--R
+--R (5) 0
+--R Type: Expression
Integer
+--E 51
+@
+
+\section{\cite{1}:14.72~~~~~$\displaystyle\int{\frac{dx}{x^3~(ax+b)^2}}$}
+$$\int{\frac{dx}{x^3~(ax+b)^2}}=
+-\frac{(ax+b)^2}{2b^4x^2}+\frac{3a(ax+b)}{b^4x}-
+\frac{a^3x}{b^4(ax+b)}-\frac{3a^2}{b^4}~\ln\left(\frac{ax+b}{x}\right)$$
+<<*>>=
+)clear all
+
+--S 52
+aa:=integrate(1/(x^3*(a*x+b)^2),x)
+--R
+--R (1)
+--R 3 3 2 2 3 3 2 2 2 2
+--R (- 6a x - 6a b x )log(a x + b) + (6a x + 6a b x )log(x) + 6a b x
+--R +
+--R 2 3
+--R 3a b x - b
+--R /
+--R 4 3 5 2
+--R 2a b x + 2b x
+--R Type: Union(Expression
Integer,...)
+--E 52
+@
+
+<<*>>=
+--S 53
+bb:=-(a*x+b)^2/(2*b^4*x^2)+(3*a*(a*x+b))/(b^4*x)-(a^3*x)/(b^4*(a*x+b))-((3*a^2)/b^4)*log((a*x+b)/x)
+--R
+--R 3 3 2 2 a x + b 3 3 2 2 2 3
+--R (- 6a x - 6a b x )log(-------) + 3a x + 9a b x + 3a b x - b
+--R x
+--R (2) ---------------------------------------------------------------
+--R 4 3 5 2
+--R 2a b x + 2b x
+--R Type: Expression
Integer
+--E 53
+@
+
+<<*>>=
+--S 54
+divlog:=rule(log(a/b) == log(a) - log(b))
+--R
+--R a
+--R (3) log(-) == - log(b) + log(a)
+--R b
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E 54
+@
+
+<<*>>=
+--S 55
+cc:=divlog bb
+--R
+--R (4)
+--R 3 3 2 2 3 3 2 2 3 3
+--R (- 6a x - 6a b x )log(a x + b) + (6a x + 6a b x )log(x) + 3a x
+--R +
+--R 2 2 2 3
+--R 9a b x + 3a b x - b
+--R /
+--R 4 3 5 2
+--R 2a b x + 2b x
+--R Type: Expression
Integer
+--E 55
+@
+
+<<*>>=
+--S 56
+cc-aa
+--R
+--R 2
+--R 3a
+--R (5) ---
+--R 4
+--R 2b
+--R Type: Expression
Integer
+--E 56
+@
+
+<<*>>=
+--S 57
+dd:=D(aa,x)
+--R
+--R 1
+--R (6) ---------------------
+--R 2 5 4 2 3
+--R a x + 2a b x + b x
+--R Type: Expression
Integer
+--E 57
+@
+
+<<*>>=
+--S 58
+ee:=D(bb,x)
+--R
+--R 1
+--R (7) ---------------------
+--R 2 5 4 2 3
+--R a x + 2a b x + b x
+--R Type: Expression
Integer
+--E 58
+@
+
+<<*>>=
+--S 59
+dd-ee
+--R
+--R (8) 0
+--R Type: Expression
Integer
+--E 59
+@
+
+\section{\cite{1}:14.73~~~~~$\displaystyle\int{\frac{dx}{(ax+b)^3}}$}
+$$\int{\frac{dx}{(ax+b)^3}}=\frac{-1}{2a(ax+b)^2}$$
+<<*>>=
+)clear all
+
+--S 60
+aa:=integrate(1/(a*x+b)^3,x)
+--R
+--R 1
+--R (1) - ----------------------
+--R 3 2 2 2
+--R 2a x + 4a b x + 2a b
+--R Type: Union(Expression
Integer,...)
+--E 60
+@
+
+{\bf NOTE: }There is a missing factor of $1/a$ in the published book.
+This factor has been inserted here.
+<<*>>=
+--S 61
+bb:=-1/(2*a*(a*x+b)^2)
+--R
+--R 1
+--R (2) - ----------------------
+--R 3 2 2 2
+--R 2a x + 4a b x + 2a b
+--R Type: Fraction Polynomial
Integer
+--E 61
+@
+
+<<*>>=
+--S 62
+aa-bb
+--R
+--R (3) 0
+--R Type: Expression
Integer
+--E 62
+@
+
+\section{\cite{1}:14.74~~~~~$\displaystyle\int{\frac{x~dx}{(ax+b)^3}}$}
+$$\int{\frac{x~dx}{(ax+b)^3}}=
+\frac{-1}{a^2(ax+b)}+\frac{b}{2a^2(ax+b)^2}$$
+<<*>>=
+)clear all
+
+--S 63
+aa:=integrate(x/(a*x+b)^3,x)
+--R
+--R - 2a x - b
+--R (1) ----------------------
+--R 4 2 3 2 2
+--R 2a x + 4a b x + 2a b
+--R Type: Union(Expression
Integer,...)
+--E 63
+@
+
+<<*>>=
+--S 64
+bb:=-1/(a^2*(a*x+b))+b/(2*a^2*(a*x+b)^2)
+--R
+--R - 2a x - b
+--R (2) ----------------------
+--R 4 2 3 2 2
+--R 2a x + 4a b x + 2a b
+--R Type: Fraction Polynomial
Integer
+--E 64
+@
+
+<<*>>=
+--S 65
+aa-bb
+--R
+--R (3) 0
+--R Type: Expression
Integer
+--E 65
+@
+
+\section{\cite{1}:14.75~~~~~$\displaystyle\int{\frac{x^2~dx}{(ax+b)^3}}$}
+$$\int{\frac{x^2~dx}{(ax+b)^3}}=
+\frac{2b}{a^3(ax+b)}-\frac{b^2}{2a^3(ax+b)^2}+
+\frac{1}{a^3}~\ln(ax+b)$$
+<<*>>=
+)clear all
+
+--S 66
+aa:=integrate(x^2/(a*x+b)^3,x)
+--R
+--R 2 2 2 2
+--R (2a x + 4a b x + 2b )log(a x + b) + 4a b x + 3b
+--R (1) -------------------------------------------------
+--R 5 2 4 3 2
+--R 2a x + 4a b x + 2a b
+--R Type: Union(Expression
Integer,...)
+--E 66
+@
+
+<<*>>=
+--S 67
+bb:=(2*b)/(a^3*(a*x+b))-(b^2)/(2*a^3*(a*x+b)^2)+1/a^3*log(a*x+b)
+--R
+--R 2 2 2 2
+--R (2a x + 4a b x + 2b )log(a x + b) + 4a b x + 3b
+--R (2) -------------------------------------------------
+--R 5 2 4 3 2
+--R 2a x + 4a b x + 2a b
+--R Type: Expression
Integer
+--E 67
+@
+
+<<*>>=
+--S 68
+aa-bb
+--R
+--R (3) 0
+--R Type: Expression
Integer
+--E 68
+@
+
+\section{\cite{1}:14.76~~~~~$\displaystyle\int{\frac{x^3~dx}{(ax+b)^3}}$}
+$$\int{\frac{x^3~dx}{(ax+b)^3}}=
+\frac{x}{a^3}-\frac{3b^2}{a^4(ax+b)}+\frac{b^3}{2a^4(ax+b)^2}-
+\frac{3b}{a^4}~\ln(ax+b)$$
+<<*>>=
+)clear all
+--S 69
+aa:=integrate(x^3/(a*x+b)^3,x)
+--R
+--R (1)
+--R 2 2 2 3 3 3 2 2 2 3
+--R (- 6a b x - 12a b x - 6b )log(a x + b) + 2a x + 4a b x - 4a b x - 5b
+--R ------------------------------------------------------------------------
+--R 6 2 5 4 2
+--R 2a x + 4a b x + 2a b
+--R Type: Union(Expression
Integer,...)
+--E 69
+@
+
+<<*>>=
+--S 70
+bb:=(x/a^3)-(3*b^2)/(a^4*(a*x+b))+b^3/(2*a^4*(a*x+b)^2)-(3*b)/a^4*log(a*x+b)
+--R
+--R (2)
+--R 2 2 2 3 3 3 2 2 2 3
+--R (- 6a b x - 12a b x - 6b )log(a x + b) + 2a x + 4a b x - 4a b x - 5b
+--R ------------------------------------------------------------------------
+--R 6 2 5 4 2
+--R 2a x + 4a b x + 2a b
+--R Type: Expression
Integer
+--E 70
+@
+
+<<*>>=
+--S 71
+aa-bb
+--R
+--R (3) 0
+--R Type: Expression
Integer
+--E 71
+@
+
+\section{\cite{1}:14.77~~~~~$\displaystyle\int{\frac{dx}{x(ax+b)^3}}$}
+$$\int{\frac{dx}{x(ax+b)^3}}=
+\frac{3}{2b(ax+b)^2}+\frac{2ax}{2b^2(ax+b)^2}-
+\frac{1}{b^3}*\ln\left(\frac{ax+b}{x}\right)$$
+
+{\bf NOTE: }The equation given in the book is wrong. This is correct.
+
+<<*>>=
+)clear all
+
+--S 72
+aa:=integrate(1/(x*(a*x+b)^3),x)
+--R
+--R (1)
+--R 2 2 2 2 2 2
+--R (- 2a x - 4a b x - 2b )log(a x + b) + (2a x + 4a b x + 2b )log(x)
+--R +
+--R 2
+--R 2a b x + 3b
+--R /
+--R 2 3 2 4 5
+--R 2a b x + 4a b x + 2b
+--R Type: Union(Expression
Integer,...)
+--E 72
+@
+
+<<*>>=
+--S 73
+bb:=3/(2*b*(a*x+b)^2)+(2*a*x)/(2*b^2*(a*x+b)^2)-1/b^3*log((a*x+b)/x)
+--R
+--R 2 2 2 a x + b 2
+--R (- 2a x - 4a b x - 2b )log(-------) + 2a b x + 3b
+--R x
+--R (2) ---------------------------------------------------
+--R 2 3 2 4 5
+--R 2a b x + 4a b x + 2b
+--R Type: Expression
Integer
+--E 73
+@
+
+<<*>>=
+--S 74
+divlog:=rule(log(a/b) == log(a) - log(b))
+--R
+--R a
+--R (3) log(-) == - log(b) + log(a)
+--R b
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E 74
+@
+
+<<*>>=
+--S 75
+cc:=divlog bb
+--R
+--R (4)
+--R 2 2 2 2 2 2
+--R (- 2a x - 4a b x - 2b )log(a x + b) + (2a x + 4a b x + 2b )log(x)
+--R +
+--R 2
+--R 2a b x + 3b
+--R /
+--R 2 3 2 4 5
+--R 2a b x + 4a b x + 2b
+--R Type: Expression
Integer
+--E 75
+@
+
+<<*>>=
+--S 76
+aa-cc
+--R
+--R (5) 0
+--R Type: Expression
Integer
+--E 76
+@
+
+\section{\cite{1}:14.78~~~~~$\displaystyle\int{\frac{dx}{x^2(ax+b)^3}}$}
+$$\int{\frac{dx}{x^2(ax+b)^3}}=
+\frac{-a}{2b^2(ax+b)^2}-\frac{2a}{b^3(ax+b)}-
+\frac{1}{b^3x}+\frac{3a}{b^4}~\ln\left(\frac{ax+b}{x}\right)$$
+<<*>>=
+)clear all
+
+--S 77
+aa:=integrate(1/(x^2*(a*x+b)^3),x)
+--R
+--R (1)
+--R 3 3 2 2 2
+--R (6a x + 12a b x + 6a b x)log(a x + b)
+--R +
+--R 3 3 2 2 2 2 2 2 3
+--R (- 6a x - 12a b x - 6a b x)log(x) - 6a b x - 9a b x - 2b
+--R /
+--R 2 4 3 5 2 6
+--R 2a b x + 4a b x + 2b x
+--R Type: Union(Expression
Integer,...)
+--E 77
+@
+
+<<*>>=
+--S 78
+bb:=-a/(2*b^2*(a*x+b)^2)-(2*a)/(b^3*(a*x+b))-1/(b^3*x)+((3*a)/b^4)*log((a*x+b)/x)
+--R
+--R 3 3 2 2 2 a x + b 2 2 2 3
+--R (6a x + 12a b x + 6a b x)log(-------) - 6a b x - 9a b x - 2b
+--R x
+--R (2) ----------------------------------------------------------------
+--R 2 4 3 5 2 6
+--R 2a b x + 4a b x + 2b x
+--R Type: Expression
Integer
+--E 78
+@
+
+<<*>>=
+--S 79
+divlog:=rule(log(a/b) == log(a) - log(b))
+--R
+--R a
+--R (3) log(-) == - log(b) + log(a)
+--R b
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E 79
+@
+
+<<*>>=
+--S 80
+cc:=divlog bb
+--R
+--R (4)
+--R 3 3 2 2 2
+--R (6a x + 12a b x + 6a b x)log(a x + b)
+--R +
+--R 3 3 2 2 2 2 2 2 3
+--R (- 6a x - 12a b x - 6a b x)log(x) - 6a b x - 9a b x - 2b
+--R /
+--R 2 4 3 5 2 6
+--R 2a b x + 4a b x + 2b x
+--R Type: Expression
Integer
+--E 80
+@
+
+<<*>>=
+--S 81
+cc-aa
+--R
+--R (5) 0
+--R Type: Expression
Integer
+--E 81
+@
+
+\section{\cite{1}:14.79~~~~~$\displaystyle\int{\frac{dx}{x^3(ax+b)^3}}$}
+$$\int{\frac{dx}{x^3(ax+b)^3}}=$$
+$$-\frac{1}{2bx^2(ax+b)^2}+
+\frac{2a}{b^2x(ax+b)^2}+
+\frac{9a^2}{b^3(ax+b)^2}+
+\frac{6a^3x}{b^4(ax+b)^2}-
+\frac{6a^2}{b^5}~\ln\left(\frac{ax+b}{x}\right)$$
+
+{\bf NOTE: }The equation given in the book is wrong. This is correct.
+
+<<*>>=
+)clear all
+
+--S 82
+aa:=integrate(1/(x^3*(a*x+b)^3),x)
+--R
+--R (1)
+--R 4 4 3 3 2 2 2
+--R (- 12a x - 24a b x - 12a b x )log(a x + b)
+--R +
+--R 4 4 3 3 2 2 2 3 3 2 2 2 3
4
+--R (12a x + 24a b x + 12a b x )log(x) + 12a b x + 18a b x + 4a b x
- b
+--R /
+--R 2 5 4 6 3 7 2
+--R 2a b x + 4a b x + 2b x
+--R Type: Union(Expression
Integer,...)
+--E 82
+@
+
+<<*>>=
+--S 83
+bb:=-1/(2*b*x^2*(a*x+b)^2)_
+ +(2*a)/(b^2*x*(a*x+b)^2)_
+ +(9*a^2)/(b^3*(a*x+b)^2)_
+ +(6*a^3*x)/(b^4*(a*x+b)^2)_
+ +(-6*a^2)/b^5*log((a*x+b)/x)
+--R
+--R (2)
+--R 4 4 3 3 2 2 2 a x + b 3 3 2 2 2
+--R (- 12a x - 24a b x - 12a b x )log(-------) + 12a b x + 18a b x
+--R x
+--R +
+--R 3 4
+--R 4a b x - b
+--R /
+--R 2 5 4 6 3 7 2
+--R 2a b x + 4a b x + 2b x
+--R Type: Expression
Integer
+--E 83
+@
+<<*>>=
+--S 84
+cc:=aa-bb
+--R
+--R 2 2 2 a x + b
+--R - 6a log(a x + b) + 6a log(x) + 6a log(-------)
+--R x
+--R (3) -----------------------------------------------
+--R 5
+--R b
+--R Type: Expression
Integer
+--E 84
+@
+
+<<*>>=
+--S 85
+divlog:=rule(log(a/b) == log(a) - log(b))
+--R
+--R a
+--R (4) log(-) == - log(b) + log(a)
+--R b
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E 85
+@
+
+<<*>>=
+--S 86
+divlog cc
+--R
+--R (5) 0
+--R Type: Expression
Integer
+--E 86
+@
+
+\section{\cite{1}:14.80~~~~~$\displaystyle\int{(ax+b)^n~dx}$}
+$$\int{(ax+b)^n~dx}=
+\frac{(ax+b)^{n+1}}{(n+1)a}{\rm\ provided\ }n \ne -1$$
+<<*>>=
+)clear all
+--S 87
+aa:=integrate((a*x+b)^n,x)
+--R
+--R n log(a x + b)
+--R (a x + b)%e
+--R (1) -------------------------
+--R a n + a
+--R Type: Union(Expression
Integer,...)
+--E 87
+@
+
+<<*>>=
+--S 88
+explog:=rule(%e^(n*log(x)) == x^n)
+--R
+--R n log(x) n
+--R (2) %e == x
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E 88
+@
+
+<<*>>=
+--S 89
+explog aa
+--R
+--R n
+--R (a x + b)(a x + b)
+--R (3) -------------------
+--R a n + a
+--R Type: Expression
Integer
+--E 89
+@
+
+\section{\cite{1}:14.81~~~~~$\displaystyle\int{x(ax+b)^n~dx}$}
+$$\int{x(ax+b)^n~dx}=
+\frac{(ax+b)^{n+2}}{(n+2)a^2}-\frac{b(ax+b)^{n+1}}{(n+1)a^2}
+{\rm\ provided\ }n \ne -1,-2$$
+
+\section{\cite{1}:14.82~~~~~$\displaystyle\int{x^2(ax+b)^n~dx}$}
+$$\int{x^2(ax+b)^n~dx}=
+\frac{(ax+b)^{n+2}}{(n+3)a^3}-
+\frac{2b(ax+b)^{n+2}}{(n+2)a^3}+
+\frac{b^2(ax+b)^{n+1}}{(n+1)a^3}
+{\rm\ provided\ }n \ne -1,-2,-3$$
+
+<<*>>=
+)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 pp60-61
+\end{thebibliography}
+\end{document}
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