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[Axiom-developer] 20080415.01.tpd.patch (CATS Schaums-Axiom equivalence
|
From: |
daly |
|
Subject: |
[Axiom-developer] 20080415.01.tpd.patch (CATS Schaums-Axiom equivalence testing (1)) |
|
Date: |
Tue, 15 Apr 2008 02:25:15 -0500 |
This patch covers schaum1.input.pamphlet which are integrals involving
a*x+b. We attempt to determine if the results from Schaums and Axiom
are equal. If the results differ by a constant, the constant is
determined. The detailed results are:
14:59 Schaums and Axiom agree
14:60 Schaums and Axiom agree
14:61 Schaums and Axiom differ by a constant
14:62 Schaums and Axiom differ by a constant
14:63 Schaums and Axiom agree
14:64 Schaums and Axiom agree
14:65 Schaums and Axiom agree
14:66 Schaums and Axiom agree
14:67 Schaums and Axiom agree
14:68 Schaums and Axiom differ by a constant
14:69 Schaums and Axiom differ by a constant
14:70 Schaums and Axiom agree
14:71 Schaums and Axiom agree
14:72 Schaums and Axiom differ by a constant
14:73 Schaums and Axiom differ by a constant
14:74 Schaums and Axiom agree
14:75 Schaums and Axiom agree
14:76 Schaums and Axiom agree
14:77 Schaums and Axiom differ by a constant
14:78 Schaums and Axiom agree
14:79 Schaums and Axiom agree
14:80 Schaums and Axiom agree
14:81 Schaums and Axiom agreement cannot be determined
14:82 Schaums and Axiom agreement cannot be determined
14:83 Axiom cannot do this integration
This has uncovered one obvious weakness in Axiom related to
indeterminants, shown in 14:81 and 14:82
Tim
==================================================================
diff --git a/changelog b/changelog
index d3ccbf9..346e9f2 100644
--- a/changelog
+++ b/changelog
@@ -1,3 +1,4 @@
+20080415 tpd src/input/schaum1.input show Schaums-Axiom equivalence
20080414 tpd src/input/Makefile add integration regression testing
20080414 tpd src/input/schaum34.input integrals of csch(ax)
20080414 tpd src/input/schaum33.input integrals of csch(ax)
diff --git a/src/input/schaum1.input.pamphlet b/src/input/schaum1.input.pamphlet
index 8507428..7e7e8c4 100644
--- a/src/input/schaum1.input.pamphlet
+++ b/src/input/schaum1.input.pamphlet
@@ -7,8 +7,11 @@
\eject
\tableofcontents
\eject
-\section{\cite{1}:14.59~~~~~$\displaystyle\int{\frac{dx}{ax+b}~dx}$}
-$$\int{\frac{dx}{ax+b}~dx}==\frac{1}{a}~\ln(ax+b)$$
+\section{\cite{1}:14.59~~~~~$\displaystyle
+\int{\frac{dx}{ax+b}}$}
+$$\int{\frac{1}{ax+b}}=
+\frac{1}{a}~\ln(ax+b)
+$$
<<*>>=
)spool schaum1.output
)set message test on
@@ -16,21 +19,40 @@ $$\int{\frac{dx}{ax+b}~dx}==\frac{1}{a}~\ln(ax+b)$$
)clear all
--S 1
-integrate(1/(a*x+b),x)
+aa:=integrate(1/(a*x+b),x)
--R
--R log(a x + b)
--R (1) ------------
--R a
--R Type: Union(Expression
Integer,...)
--E 1
+
+--S 2
+bb:=1/a*log(a*x+b)
+--R
+--R log(a x + b)
+--R (2) ------------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 3 14:59 Schaums and Axiom agree
+cc:=bb-aa
+--R
+--R (3) 0
+--R Type: Expression
Integer
+--E
@
-\section{\cite{1}:14.60~~~~~$\displaystyle\int{\frac{x~dx}{ax+b}}$}
-$$\int{\frac{x~dx}{ax+b}}=\frac{x}{a}-\frac{b}{a^2}~\ln(ax+b)$$
+\section{\cite{1}:14.60~~~~~$\displaystyle
+\int{\frac{x~dx}{ax+b}}$}
+$$\int{\frac{x}{ax+b}}=
+\frac{x}{a}-\frac{b}{a^2}~\ln(ax+b)
+$$
<<*>>=
)clear all
---S 2
-integrate(x/(a*x+b),x)
+--S 4
+aa:=integrate(x/(a*x+b),x)
--R
--R
--R - b log(a x + b) + a x
@@ -38,16 +60,36 @@ integrate(x/(a*x+b),x)
--R 2
--R a
--R Type: Union(Expression
Integer,...)
---E 2
+--E
+
+--S 5
+bb:=x/a-b/a^2*log(a*x+b)
+--R
+--R - b log(a x + b) + a x
+--R (2) ----------------------
+--R 2
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 6 14:60 Schaums and Axiom agree
+cc:=bb-aa
+--R
+--R (3) 0
+--R Type: Expression
Integer
+--E
@
-\section{\cite{1}:14.61~~~~~$\displaystyle\int{\frac{x^2~dx}{ax+b}}$}
-$$\int{\frac{x^2~dx}{ax+b}}=
-\frac{(ax+b)^2}{2a^3}-\frac{2b(ax+b)}{a^3}+\frac{b^2}{a^3}~\ln(ax+b)$$
+
+\section{\cite{1}:14.61~~~~~$\displaystyle
+\int{\frac{x^2~dx}{ax+b}}$}
+$$\int{\frac{x^2}{ax+b}}=
+\frac{(ax+b)^2}{2a^3}-\frac{2b(ax+b)}{a^3}+\frac{b^2}{a^3}~\ln(ax+b)
+$$
<<*>>=
)clear all
---S 3
-nn:=integrate(x^2/(a*x+b),x)
+--S 7
+aa:=integrate(x^2/(a*x+b),x)
--R
--R 2 2 2
--R 2b log(a x + b) + a x - 2a b x
@@ -55,13 +97,10 @@ nn:=integrate(x^2/(a*x+b),x)
--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)
+--E
+
+--S 8
+bb:=(a*x+b)^2/(2*a^3)-(2*b*(a*x+b))/a^3+b^2/a^3*log(a*x+b)
--R
--R 2 2 2 2
--R 2b log(a x + b) + a x - 2a b x - 3b
@@ -69,12 +108,10 @@ mm:=((a*x+b)^2-2*2*b*(a*x+b)+2*b^2*log(a*x+b))/(2*a^3)
--R 3
--R 2a
--R Type: Expression
Integer
---E 4
-@
-and we take their difference:
-<<*>>=
---S 5
-pp:=mm-nn
+--E
+
+--S 9
+cc:=bb-aa
--R
--R 2
--R 3b
@@ -82,50 +119,28 @@ pp:=mm-nn
--R 3
--R 2a
--R Type: Expression
Integer
---E 5
+--E
@
-which is a constant with respect to x, and thus the constant C.
+This factor is constant with respect to $x$ as shown by taking the
+derivative. It is a constant of integration.
<<*>>=
---S 6
-D(pp,x)
+--S 10 14:61 Schaums and Axiom differ by a constant
+differentiate(cc,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
+--E
@
-<<*>>=
---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~~~~~$\displaystyle\int{\frac{x^3~dx}{ax+b}}$}
-$$\int{\frac{x^3~dx}{ax+b}}=
+\section{\cite{1}:14.62~~~~~$\displaystyle
+\int{\frac{x^3~dx}{ax+b}}$}
+$$\int{\frac{x^3}{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)$$
+\frac{3b^2(ax+b)}{a^4}-\frac{b^3}{a^4}~\ln(ax+b)
+$$
<<*>>=
)clear all
---S 9
+--S 11
aa:=integrate(x^3/(a*x+b),x)
--R
--R 3 3 3 2 2 2
@@ -134,11 +149,11 @@ aa:=integrate(x^3/(a*x+b),x)
--R 4
--R 6a
--R Type: Union(Expression
Integer,...)
---E 9
+--E
@
and the book expression is:
<<*>>=
---S 10
+--S 12
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
@@ -147,13 +162,13 @@
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(
--R 4
--R 6a
--R Type: Expression
Integer
---E 10
+--E
@
The difference is a constant with respect to x:
<<*>>=
---S 11
-aa-bb
+--S 13
+cc:=aa-bb
--R
--R 3
--R 11b
@@ -161,90 +176,92 @@ aa-bb
--R 4
--R 6a
--R Type: Expression
Integer
---E 11
+--E
@
-If we differentiate each expression we see
+If we differentiate each expression we see that this is the integration
+constant.
<<*>>=
---S 12
-cc:=D(aa,x)
+--S 14 14:62 Schaums and Axiom differ by a constant
+dd:=D(cc,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 (4) 0
--R Type: Expression
Integer
---E 14
+--E
@
-\section{\cite{1}:14.63~~~~~$\displaystyle\int{\frac{dx}{x~(ax+b)}}$}
-$$\int{\frac{dx}{x~(ax+b)}}=\frac{1}{b}~\ln\left(\frac{x}{ax+b}\right)$$
+\section{\cite{1}:14.63~~~~~$\displaystyle
+\int{\frac{dx}{x~(ax+b)}}$}
+$$\int{\frac{1}{x~(ax+b)}}=
+\frac{1}{b}~\ln\left(\frac{x}{ax+b}\right)
+$$
<<*>>=
)clear all
--S 15
-ff:=integrate(1/(x*(a*x+b)),x)
+aa:=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
+--E
+
+--S 16
+bb:=1/b*log(x/(a*x+b))
+--R
+--R x
+--R log(-------)
+--R a x + b
+--R (2) ------------
+--R b
+--R Type: Expression
Integer
+--E
+
+--S 17
+cc:=aa-bb
+--R
+--R x
+--R - log(a x + b) + log(x) - log(-------)
+--R a x + b
+--R (3) --------------------------------------
+--R b
+--R Type: Expression
Integer
+--E
@
but we know that $$\log(a)-\log(b)=\log(\frac{a}{b})$$
We can express this fact as a rule:
<<*>>=
---S 16
+--S 18
logdiv:=rule(log(a)-log(b) == log(a/b))
--R
--R a
---I (2) - log(b) + log(a) + %I == log(-) + %I
+--I (4) - log(b) + log(a) + %I == log(-) + %I
--R b
--R Type: RewriteRule(Integer,Integer,Expression
Integer)
---E 16
+--E
@
and use this rule to rewrite the logs into divisions:
<<*>>=
---S 17
-logdiv ff
+--S 19 14:63 Schaums and Axiom agree
+dd:=logdiv cc
--R
---R x
---R log(-------)
---R a x + b
---R (3) ------------
---R b
+--R (5) 0
--R Type: Expression
Integer
---E 17
+--E
@
so we can see the equivalence directly.
-\section{\cite{1}:14.64~~~~~$\displaystyle\int{\frac{dx}{x^2~(ax+b)}}$}
-$$\int{\frac{dx}{x^2~(ax+b)}}=
--\frac{1}{bx}+\frac{a}{b^2}~\ln\left(\frac{ax+b}{x}\right)$$
+\section{\cite{1}:14.64~~~~~$\displaystyle
+\int{\frac{dx}{x^2~(ax+b)}}$}
+$$\int{\frac{1}{x^2~(ax+b)}}=
+-\frac{1}{bx}+\frac{a}{b^2}~\ln\left(\frac{ax+b}{x}\right)
+$$
<<*>>=
)clear all
---S 18
+--S 20
aa:=integrate(1/(x^2*(a*x+b)),x)
--R
--R a x log(a x + b) - a x log(x) - b
@@ -252,12 +269,12 @@ aa:=integrate(1/(x^2*(a*x+b)),x)
--R 2
--R b x
--R Type: Union(Expression
Integer,...)
---E 18
+--E
@
The original form given in the book expands to:
<<*>>=
---S 19
+--S 21
bb:=-1/(b*x)+a/b^2*log((a*x+b)/x)
--R
--R a x + b
@@ -267,48 +284,50 @@ bb:=-1/(b*x)+a/b^2*log((a*x+b)/x)
--R 2
--R b x
--R Type: Expression
Integer
---E 19
+--E
+
+--S 22
+cc:=aa-bb
+--R
+--R a x + b
+--R a log(a x + b) - a log(x) - a log(-------)
+--R x
+--R (3) ------------------------------------------
+--R 2
+--R b
+--R Type: Expression
Integer
+--E
@
We can define the following rule to expand log forms:
<<*>>=
---S 20
+--S 23
divlog:=rule(log(a/b) == log(a) - log(b))
--R
--R a
---R (3) log(-) == - log(b) + log(a)
+--R (4) 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
+--E
@
-and we can now see that the results are identical.
+and apply it to the difference
<<*>>=
---S 22
-aa-cc
+--S 24 14:64 Schaums and Axiom agree
+divlog cc
--R
--R (5) 0
--R Type: Expression
Integer
---E 22
+--E
@
-\section{\cite{1}:14.65~~~~~$\displaystyle\int{\frac{dx}{x^3~(ax+b)}}$}
-$$\int{\frac{dx}{x^3~(ax+b)}}=
-\frac{2ax-b}{2b^2x^2}+\frac{a^2}{b^3}~\ln\left(\frac{x}{ax+b}\right)$$
+\section{\cite{1}:14.65~~~~~$\displaystyle
+\int{\frac{dx}{x^3~(ax+b)}}$}
+$$\int{\frac{1}{x^3~(ax+b)}}=
+\frac{2ax-b}{2b^2x^2}+\frac{a^2}{b^3}~\ln\left(\frac{x}{ax+b}\right)
+$$
<<*>>=
)clear all
---S 23
+--S 25
aa:=integrate(1/(x^3*(a*x+b)),x)
--R
--R 2 2 2 2 2
@@ -317,11 +336,9 @@ aa:=integrate(1/(x^3*(a*x+b)),x)
--R 3 2
--R 2b x
--R Type: Union(Expression
Integer,...)
---E 23
-@
+--E
-<<*>>=
---S 24
+--S 26
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
@@ -331,95 +348,121 @@ bb:=(2*a*x-b)/(2*b^2*x^2)+a^2/b^3*log(x/(a*x+b))
--R 3 2
--R 2b x
--R Type: Expression
Integer
---E 24
-@
+--E
-<<*>>=
---S 25
+--S 27
+cc:=aa-bb
+--R
+--R 2 2 2 x
+--R - a log(a x + b) + a log(x) - a log(-------)
+--R a x + b
+--R (3) --------------------------------------------
+--R 3
+--R b
+--R Type: Expression
Integer
+--E
+
+--S 28
divlog:=rule(log(a/b) == log(a) - log(b))
--R
--R a
---R (3) log(-) == - log(b) + log(a)
+--R (4) log(-) == - log(b) + log(a)
--R b
--R Type: RewriteRule(Integer,Integer,Expression
Integer)
---E 25
-@
+--E
-<<*>>=
---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
+--S 29 14:65 Schaums and Axiom agree
+dd:=divlog cc
--R
--R (5) 0
--R Type: Expression
Integer
---E 27
+--E
@
-\section{\cite{1}:14.66~~~~~$\displaystyle\int{\frac{dx}{(ax+b)^2}}$}
-$$\int{\frac{dx}{(ax+b)^2}}=\frac{-1}{a~(ax+b)}$$
+\section{\cite{1}:14.66~~~~~$\displaystyle
+\int{\frac{dx}{(ax+b)^2}}$}
+$$\int{\frac{1}{(ax+b)^2}}=
+\frac{-1}{a~(ax+b)}
+$$
<<*>>=
)clear all
---S 28
-integrate(1/(a*x+b)^2,x)
+--S 30
+aa:=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
+--E
+
+--S 31
+bb:=-1/(a*(a*x+b))
+--R
+--R 1
+--R (2) - ---------
+--R 2
+--R a x + a b
+--R Type: Fraction Polynomial
Integer
+--E
+
+--S 32 14:66 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression
Integer
+--E
+
@
-\section{\cite{1}:14.67~~~~~$\displaystyle\int{\frac{x~dx}{(ax+b)^2}}$}
-$$\int{\frac{x~dx}{(ax+b)^2}}=
-\frac{b}{a^2~(ax+b)}+\frac{1}{a^2}~\ln(ax+b)$$
+\section{\cite{1}:14.67~~~~~$\displaystyle
+\int{\frac{x~dx}{(ax+b)^2}}$}
+$$\int{\frac{x}{(ax+b)^2}}=
+\frac{b}{a^2~(ax+b)}+\frac{1}{a^2}~\ln(ax+b)
+$$
<<*>>=
)clear all
---S 29
-integrate(x/(a*x+b)^2,x)
+--S 33
+aa:=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)
+--E
+
+--S 34
+bb:=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
+--E
+
+--S 35 14:67 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression
Integer
+--E
+
@
-\section{\cite{1}:14.68~~~~~$\displaystyle\int{\frac{x^2~dx}{(ax+b)^2}}$}
-$$\int{\frac{x^2~dx}{(ax+b)^2}}=
+\section{\cite{1}:14.68~~~~~$\displaystyle
+\int{\frac{x^2~dx}{(ax+b)^2}}$}
+$$\int{\frac{x^2}{(ax+b)^2}}=
\frac{ax+b}{a^3}-\frac{b^2}{a^3~(ax+b)}
--\frac{2b}{a^3}~\ln(ax+b)$$
+-\frac{2b}{a^3}~\ln(ax+b)
+$$
<<*>>=
)clear all
---S 31
+--S 36
aa:=integrate(x^2/(a*x+b)^2,x)
--R
--R 2 2 2 2
@@ -428,11 +471,11 @@ aa:=integrate(x^2/(a*x+b)^2,x)
--R 4 3
--R a x + a b
--R Type: Union(Expression
Integer,...)
---E 31
+--E
@
and the book expression expands into
<<*>>=
---S 32
+--S 37
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
@@ -441,57 +484,42 @@ bb:=(a*x+b)/a^3-b^2/(a^3*(a*x+b))-((2*b)/a^3)*log(a*x+b)
--R 4 3
--R a x + a b
--R Type: Expression
Integer
---E 32
+--E
@
These two expressions differ by the constant
<<*>>=
---S 33
-aa-bb
+--S 38
+cc:=aa-bb
--R
--R b
--R (3) - --
--R 3
--R a
--R Type: Expression
Integer
---E 33
+--E
@
-These are the same integrands as can be shown by differentiation:
+That this expression is constant can be shown by differentiation:
<<*>>=
---S 34
-D(aa,x)
+--S 39 14:68 Schaums and Axiom differ by a constant
+D(cc,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 (4) 0
--R Type: Expression
Integer
---E 35
+--E
@
-\section{\cite{1}:14.69~~~~~$\displaystyle\int{\frac{x^3~dx}{(ax+b)^2}}$}
-$$\int{\frac{x^3~dx}{(ax+b)^2}}=
+\section{\cite{1}:14.69~~~~~$\displaystyle
+\int{\frac{x^3~dx}{(ax+b)^2}}$}
+$$\int{\frac{x^3}{(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)$$
++\frac{3b^2}{a^4}~\ln(ax+b)
+$$
<<*>>=
)clear all
---S 36
+--S 40
aa:=integrate(x^3/(a*x+b)^2,x)
--R
--R 2 3 3 3 2 2 2 3
@@ -500,11 +528,9 @@ aa:=integrate(x^3/(a*x+b)^2,x)
--R 5 4
--R 2a x + 2a b
--R Type: Union(Expression
Integer,...)
---E 36
-@
+--E
-<<*>>=
---S 37
+--S 41
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
@@ -513,12 +539,10 @@
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 5 4
--R 2a x + 2a b
--R Type: Expression
Integer
---E 37
-@
+--E
-<<*>>=
---S 38
-aa-bb
+--S 42
+cc:=aa-bb
--R
--R 2
--R 5b
@@ -526,51 +550,24 @@ aa-bb
--R 4
--R 2a
--R Type: Expression
Integer
---E 38
-@
+--E
-<<*>>=
---S 39
-cc:=D(aa,x)
+--S 43 14:69 Schaums and Axiom differ by a constant
+dd:=D(cc,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 (4) 0
--R Type: Expression
Integer
---E 41
+--E
@
-
-\section{\cite{1}:14.70~~~~~$\displaystyle\int{\frac{dx}{x~(ax+b)^2}}$}
-$$\int{\frac{dx}{x~(ax+b)^2}}=
-\frac{1}{b~(ax+b)}+\frac{1}{b^2}~\ln\left(\frac{x}{ax+b}\right)$$
+\section{\cite{1}:14.70~~~~~$\displaystyle
+\int{\frac{dx}{x~(ax+b)^2}}$}
+$$\int{\frac{1}{x~(ax+b)^2}}=
+\frac{1}{b~(ax+b)}+\frac{1}{b^2}~\ln\left(\frac{x}{ax+b}\right)
+$$
<<*>>=
)clear all
---S 42
+--S 44
aa:=integrate(1/(x*(a*x+b)^2),x)
--R
--R (- a x - b)log(a x + b) + (a x + b)log(x) + b
@@ -578,11 +575,11 @@ aa:=integrate(1/(x*(a*x+b)^2),x)
--R 2 3
--R a b x + b
--R Type: Union(Expression
Integer,...)
---E 42
+--E
@
and the book says:
<<*>>=
---S 43
+--S 45
bb:=(1/(b*(a*x+b))+(1/b^2)*log(x/(a*x+b)))
--R
--R x
@@ -592,51 +589,52 @@ bb:=(1/(b*(a*x+b))+(1/b^2)*log(x/(a*x+b)))
--R 2 3
--R a b x + b
--R Type: Expression
Integer
---E 43
-@
+--E
+--S 46
+cc:=aa-bb
+--R
+--R x
+--R - log(a x + b) + log(x) - log(-------)
+--R a x + b
+--R (3) --------------------------------------
+--R 2
+--R b
+--R Type: Expression
Integer
+--E
+@
So we look at the divlog rule again:
<<*>>=
---S 44
+--S 47
divlog:=rule(log(a/b) == log(a) - log(b))
--R
--R a
---R (3) log(-) == - log(b) + log(a)
+--R (4) log(-) == - log(b) + log(a)
--R b
--R Type: RewriteRule(Integer,Integer,Expression
Integer)
---E 44
+--E
@
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
+--S 48 14:70 Schaums and Axiom agree
+dd:=divlog cc
--R
--R (5) 0
--R Type: Expression
Integer
---E 46
+--E
@
-\section{\cite{1}:14.71~~~~~$\displaystyle\int{\frac{dx}{x^2~(ax+b)^2}}$}
-$$\int{\frac{dx}{x^2~(ax+b)^2}}=
+\section{\cite{1}:14.71~~~~~$\displaystyle
+\int{\frac{dx}{x^2~(ax+b)^2}}$}
+$$\int{\frac{1}{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)$$
+\frac{2a}{b^3}~\ln\left(\frac{ax+b}{x}\right)
+$$
<<*>>=
)clear all
---S 47
+--S 49
aa:=integrate(1/(x^2*(a*x+b)^2),x)
--R
--R 2 2 2 2
2
@@ -645,11 +643,11 @@ aa:=integrate(1/(x^2*(a*x+b)^2),x)
--R 3 2 4
--R a b x + b x
--R Type: Union(Expression
Integer,...)
---E 47
+--E
@
and the book says:
<<*>>=
---S 48
+--S 50
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
@@ -659,50 +657,50 @@
bb:=(-a/(b^2*(a*x+b)))-(1/(b^2*x))+((2*a)/b^3)*log((a*x+b)/x)
--R 3 2 4
--R a b x + b x
--R Type: Expression
Integer
---E 48
+--E
+
+--S 51
+cc:=aa-bb
+--R
+--R a x + b
+--R 2a log(a x + b) - 2a log(x) - 2a log(-------)
+--R x
+--R (3) ---------------------------------------------
+--R 3
+--R b
+--R Type: Expression
Integer
+--E
@
which calls for our divlog rule:
<<*>>=
---S 49
+--S 52
divlog:=rule(log(a/b) == log(a) - log(b))
--R
--R a
---R (3) log(-) == - log(b) + log(a)
+--R (4) log(-) == - log(b) + log(a)
--R b
--R Type: RewriteRule(Integer,Integer,Expression
Integer)
---E 49
+--E
@
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
+--S 53 14:71 Schaums and Axiom agree
+dd:=divlog cc
--R
--R (5) 0
--R Type: Expression
Integer
---E 51
+--E
@
-
-\section{\cite{1}:14.72~~~~~$\displaystyle\int{\frac{dx}{x^3~(ax+b)^2}}$}
-$$\int{\frac{dx}{x^3~(ax+b)^2}}=
+\section{\cite{1}:14.72~~~~~$\displaystyle
+\int{\frac{dx}{x^3~(ax+b)^2}}$}
+$$\int{\frac{1}{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)$$
+\frac{a^3x}{b^4(ax+b)}-\frac{3a^2}{b^4}~\ln\left(\frac{ax+b}{x}\right)
+$$
<<*>>=
)clear all
---S 52
+--S 54
aa:=integrate(1/(x^3*(a*x+b)^2),x)
--R
--R (1)
@@ -715,11 +713,9 @@ aa:=integrate(1/(x^3*(a*x+b)^2),x)
--R 4 3 5 2
--R 2a b x + 2b x
--R Type: Union(Expression
Integer,...)
---E 52
-@
+--E
-<<*>>=
---S 53
+--S 55
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
@@ -729,85 +725,53 @@
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)/
--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
-@
+--E
-<<*>>=
--S 56
-cc-aa
+cc:=aa-bb
--R
---R 2
---R 3a
---R (5) ---
---R 4
---R 2b
+--R 2 2 2 a x + b 2
+--R - 6a log(a x + b) + 6a log(x) + 6a log(-------) - 3a
+--R x
+--R (3) -----------------------------------------------------
+--R 4
+--R 2b
--R Type: Expression
Integer
---E 56
-@
+--E
-<<*>>=
--S 57
-dd:=D(aa,x)
+divlog:=rule(log(a/b) == log(a) - log(b))
--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
-@
+--R a
+--R (4) log(-) == - log(b) + log(a)
+--R b
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
-<<*>>=
--S 58
-ee:=D(bb,x)
+dd:=divlog cc
--R
---R 1
---R (7) ---------------------
---R 2 5 4 2 3
---R a x + 2a b x + b x
+--R 2
+--R 3a
+--R (5) - ---
+--R 4
+--R 2b
--R Type: Expression
Integer
---E 58
-@
+--E
-<<*>>=
---S 59
-dd-ee
+--S 59 14:72 Schaums and Axiom differ by a constant
+ee:=D(dd,x)
--R
---R (8) 0
+--R (6) 0
--R Type: Expression
Integer
---E 59
+--E
@
-\section{\cite{1}:14.73~~~~~$\displaystyle\int{\frac{dx}{(ax+b)^3}}$}
-$$\int{\frac{dx}{(ax+b)^3}}=\frac{-1}{2a(ax+b)^2}$$
+\section{\cite{1}:14.73~~~~~$\displaystyle
+\int{\frac{dx}{(ax+b)^3}}$}
+$$\int{\frac{1}{(ax+b)^3}}=
+\frac{-1}{2a(ax+b)^2}
+$$
<<*>>=
)clear all
@@ -819,39 +783,54 @@ aa:=integrate(1/(a*x+b)^3,x)
--R 3 2 2 2
--R 2a x + 4a b x + 2a b
--R Type: Union(Expression
Integer,...)
---E 60
-@
+--E
-{\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)
+bb:=-1/(2*(a*x+b)^2)
--R
---R 1
---R (2) - ----------------------
---R 3 2 2 2
---R 2a x + 4a b x + 2a b
+--R 1
+--R (2) - --------------------
+--R 2 2 2
+--R 2a x + 4a b x + 2b
--R Type: Fraction Polynomial
Integer
---E 61
-@
+--E
-<<*>>=
--S 62
-aa-bb
+cc:=aa-bb
--R
---R (3) 0
+--R a - 1
+--R (3) ----------------------
+--R 3 2 2 2
+--R 2a x + 4a b x + 2a b
+--R Type: Expression
Integer
+--E
+
+--S 63
+dd:=aa/bb
+--R
+--R 1
+--R (4) -
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 64 14:73 Schaums and Axiom differ by a constant
+ee:=D(dd,x)
+--R
+--R (5) 0
--R Type: Expression
Integer
---E 62
+--E
@
-\section{\cite{1}:14.74~~~~~$\displaystyle\int{\frac{x~dx}{(ax+b)^3}}$}
-$$\int{\frac{x~dx}{(ax+b)^3}}=
-\frac{-1}{a^2(ax+b)}+\frac{b}{2a^2(ax+b)^2}$$
+\section{\cite{1}:14.74~~~~~$\displaystyle
+\int{\frac{x~dx}{(ax+b)^3}}$}
+$$\int{\frac{x}{(ax+b)^3}}=
+\frac{-1}{a^2(ax+b)}+\frac{b}{2a^2(ax+b)^2}
+$$
<<*>>=
)clear all
---S 63
+--S 65
aa:=integrate(x/(a*x+b)^3,x)
--R
--R - 2a x - b
@@ -859,11 +838,9 @@ aa:=integrate(x/(a*x+b)^3,x)
--R 4 2 3 2 2
--R 2a x + 4a b x + 2a b
--R Type: Union(Expression
Integer,...)
---E 63
-@
+--E
-<<*>>=
---S 64
+--S 66
bb:=-1/(a^2*(a*x+b))+b/(2*a^2*(a*x+b)^2)
--R
--R - 2a x - b
@@ -871,26 +848,26 @@ bb:=-1/(a^2*(a*x+b))+b/(2*a^2*(a*x+b)^2)
--R 4 2 3 2 2
--R 2a x + 4a b x + 2a b
--R Type: Fraction Polynomial
Integer
---E 64
-@
+--E
-<<*>>=
---S 65
-aa-bb
+--S 67 14:74 Schaums and Axiom agree
+cc:=aa-bb
--R
--R (3) 0
--R Type: Expression
Integer
---E 65
+--E
@
-\section{\cite{1}:14.75~~~~~$\displaystyle\int{\frac{x^2~dx}{(ax+b)^3}}$}
-$$\int{\frac{x^2~dx}{(ax+b)^3}}=
+\section{\cite{1}:14.75~~~~~$\displaystyle
+\int{\frac{x^2~dx}{(ax+b)^3}}$}
+$$\int{\frac{x^2}{(ax+b)^3}}=
\frac{2b}{a^3(ax+b)}-\frac{b^2}{2a^3(ax+b)^2}+
-\frac{1}{a^3}~\ln(ax+b)$$
+\frac{1}{a^3}~\ln(ax+b)
+$$
<<*>>=
)clear all
---S 66
+--S 68
aa:=integrate(x^2/(a*x+b)^3,x)
--R
--R 2 2 2 2
@@ -899,11 +876,9 @@ aa:=integrate(x^2/(a*x+b)^3,x)
--R 5 2 4 3 2
--R 2a x + 4a b x + 2a b
--R Type: Union(Expression
Integer,...)
---E 66
-@
+--E
-<<*>>=
---S 67
+--S 69
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
@@ -912,25 +887,25 @@
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 5 2 4 3 2
--R 2a x + 4a b x + 2a b
--R Type: Expression
Integer
---E 67
-@
+--E
-<<*>>=
---S 68
-aa-bb
+--S 70 14:75 Schaums and Axiom agree
+cc:=aa-bb
--R
--R (3) 0
--R Type: Expression
Integer
---E 68
+--E
@
-\section{\cite{1}:14.76~~~~~$\displaystyle\int{\frac{x^3~dx}{(ax+b)^3}}$}
-$$\int{\frac{x^3~dx}{(ax+b)^3}}=
+\section{\cite{1}:14.76~~~~~$\displaystyle
+\int{\frac{x^3~dx}{(ax+b)^3}}$}
+$$\int{\frac{x^3}{(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)$$
+\frac{3b}{a^4}~\ln(ax+b)
+$$
<<*>>=
)clear all
---S 69
+--S 71
aa:=integrate(x^3/(a*x+b)^3,x)
--R
--R (1)
@@ -940,11 +915,9 @@ aa:=integrate(x^3/(a*x+b)^3,x)
--R 6 2 5 4 2
--R 2a x + 4a b x + 2a b
--R Type: Union(Expression
Integer,...)
---E 69
-@
+--E
-<<*>>=
---S 70
+--S 72
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)
@@ -954,29 +927,27 @@
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 6 2 5 4 2
--R 2a x + 4a b x + 2a b
--R Type: Expression
Integer
---E 70
-@
+--E
-<<*>>=
---S 71
-aa-bb
+--S 73 14:76 Schaums and Axiom agree
+cc:=aa-bb
--R
--R (3) 0
--R Type: Expression
Integer
---E 71
+--E
@
-\section{\cite{1}:14.77~~~~~$\displaystyle\int{\frac{dx}{x(ax+b)^3}}$}
-$$\int{\frac{dx}{x(ax+b)^3}}=
+\section{\cite{1}:14.77~~~~~$\displaystyle
+\int{\frac{dx}{x(ax+b)^3}}$}
+$$\int{\frac{1}{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)$$
-
-{\bf NOTE: }The equation given in the book is wrong. This is correct.
+\frac{1}{b^3}*\ln\left(\frac{ax+b}{x}\right)
+$$
<<*>>=
)clear all
---S 72
+--S 74
aa:=integrate(1/(x*(a*x+b)^3),x)
--R
--R (1)
@@ -989,68 +960,69 @@ aa:=integrate(1/(x*(a*x+b)^3),x)
--R 2 3 2 4 5
--R 2a b x + 4a b x + 2b
--R Type: Union(Expression
Integer,...)
---E 72
-@
+--E
-<<*>>=
---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)
+--S 75
+bb:=(a^2*x^2)/(2*b^3*(a*x+b)^2)-(2*a*x)/(b^3*(a*x+b))-(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 2 2 2 a x + b 2 2
+--R (- 2a x - 4a b x - 2b )log(-------) - 3a x - 4a b x
--R x
---R (2) ---------------------------------------------------
---R 2 3 2 4 5
---R 2a b x + 4a b x + 2b
+--R (2) -----------------------------------------------------
+--R 2 3 2 4 5
+--R 2a b x + 4a b x + 2b
--R Type: Expression
Integer
---E 73
-@
+--E
-<<*>>=
---S 74
+--S 76
+cc:=aa-bb
+--R
+--R a x + b
+--R - 2log(a x + b) + 2log(x) + 2log(-------) + 3
+--R x
+--R (3) ---------------------------------------------
+--R 3
+--R 2b
+--R Type: Expression
Integer
+--E
+
+--S 77
divlog:=rule(log(a/b) == log(a) - log(b))
--R
--R a
---R (3) log(-) == - log(b) + log(a)
+--R (4) log(-) == - log(b) + log(a)
--R b
--R Type: RewriteRule(Integer,Integer,Expression
Integer)
---E 74
-@
+--E
-<<*>>=
---S 75
-cc:=divlog bb
+--S 78
+dd:=divlog cc
--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 3
+--R (5) ---
+--R 3
+--R 2b
--R Type: Expression
Integer
---E 75
-@
+--E
-<<*>>=
---S 76
-aa-cc
+--S 79 14:77 Schaums and Axiom differ by a constant
+ee:=D(dd,x)
--R
---R (5) 0
+--R (6) 0
--R Type: Expression
Integer
---E 76
+--E
@
-\section{\cite{1}:14.78~~~~~$\displaystyle\int{\frac{dx}{x^2(ax+b)^3}}$}
-$$\int{\frac{dx}{x^2(ax+b)^3}}=
+\section{\cite{1}:14.78~~~~~$\displaystyle
+\int{\frac{dx}{x^2(ax+b)^3}}$}
+$$\int{\frac{1}{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)$$
+\frac{1}{b^3x}+\frac{3a}{b^4}~\ln\left(\frac{ax+b}{x}\right)
+$$
<<*>>=
)clear all
---S 77
+--S 80
aa:=integrate(1/(x^2*(a*x+b)^3),x)
--R
--R (1)
@@ -1063,11 +1035,9 @@ aa:=integrate(1/(x^2*(a*x+b)^3),x)
--R 2 4 3 5 2 6
--R 2a b x + 4a b x + 2b x
--R Type: Union(Expression
Integer,...)
---E 77
-@
+--E
-<<*>>=
---S 78
+--S 81
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
@@ -1077,60 +1047,50 @@
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 2 4 3 5 2 6
--R 2a b x + 4a b x + 2b x
--R Type: Expression
Integer
---E 78
-@
+--E
-<<*>>=
---S 79
+--S 82
+cc:=aa-bb
+--R
+--R a x + b
+--R 3a log(a x + b) - 3a log(x) - 3a log(-------)
+--R x
+--R (3) ---------------------------------------------
+--R 4
+--R b
+--R Type: Expression
Integer
+--E
+
+--S 83
divlog:=rule(log(a/b) == log(a) - log(b))
--R
--R a
---R (3) log(-) == - log(b) + log(a)
+--R (4) log(-) == - log(b) + log(a)
--R b
--R Type: RewriteRule(Integer,Integer,Expression
Integer)
---E 79
-@
+--E
-<<*>>=
---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
+--S 84 14:78 Schaums and Axiom agree
+dd:=divlog cc
--R
--R (5) 0
--R Type: Expression
Integer
---E 81
+--E
@
-\section{\cite{1}:14.79~~~~~$\displaystyle\int{\frac{dx}{x^3(ax+b)^3}}$}
-$$\int{\frac{dx}{x^3(ax+b)^3}}=$$
-$$-\frac{1}{2bx^2(ax+b)^2}+
+\section{\cite{1}:14.79~~~~~$\displaystyle
+\int{\frac{dx}{x^3(ax+b)^3}}$}
+$$\int{\frac{1}{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)$$
-{\bf NOTE: }The equation given in the book is wrong. This is correct.
-
<<*>>=
)clear all
---S 82
+--S 85
aa:=integrate(1/(x^3*(a*x+b)^3),x)
--R
--R (1)
@@ -1143,11 +1103,9 @@ aa:=integrate(1/(x^3*(a*x+b)^3),x)
--R 2 5 4 6 3 7 2
--R 2a b x + 4a b x + 2b x
--R Type: Union(Expression
Integer,...)
---E 82
-@
+--E
-<<*>>=
---S 83
+--S 86
bb:=-1/(2*b*x^2*(a*x+b)^2)_
+(2*a)/(b^2*x*(a*x+b)^2)_
+(9*a^2)/(b^3*(a*x+b)^2)_
@@ -1165,10 +1123,9 @@ bb:=-1/(2*b*x^2*(a*x+b)^2)_
--R 2 5 4 6 3 7 2
--R 2a b x + 4a b x + 2b x
--R Type: Expression
Integer
---E 83
-@
-<<*>>=
---S 84
+--E
+
+--S 87
cc:=aa-bb
--R
--R 2 2 2 a x + b
@@ -1178,35 +1135,33 @@ cc:=aa-bb
--R 5
--R b
--R Type: Expression
Integer
---E 84
-@
+--E
-<<*>>=
---S 85
+--S 88
divlog:=rule(log(a/b) == log(a) - log(b))
--R
--R a
--R (4) log(-) == - log(b) + log(a)
--R b
--R Type: RewriteRule(Integer,Integer,Expression
Integer)
---E 85
-@
+--E
-<<*>>=
---S 86
-divlog cc
+--S 89 14:79 Schaums and Axiom agree
+dd:=divlog cc
--R
--R (5) 0
--R Type: Expression
Integer
---E 86
+--E
@
-\section{\cite{1}:14.80~~~~~$\displaystyle\int{(ax+b)^n~dx}$}
-$$\int{(ax+b)^n~dx}=
-\frac{(ax+b)^{n+1}}{(n+1)a}{\rm\ provided\ }n \ne -1$$
+\section{\cite{1}:14.80~~~~~$\displaystyle
+\int{(ax+b)^n~dx}$}
+$$\int{(ax+b)^n}=
+\frac{(ax+b)^{n+1}}{(n+1)a}{\rm\ provided\ }n \ne -1
+$$
<<*>>=
)clear all
---S 87
+--S 90
aa:=integrate((a*x+b)^n,x)
--R
--R n log(a x + b)
@@ -1214,44 +1169,208 @@ aa:=integrate((a*x+b)^n,x)
--R (1) -------------------------
--R a n + a
--R Type: Union(Expression
Integer,...)
---E 87
-@
+--E
+
+--S 91
+bb:=(a*x+b)^(n+1)/((n+1)*a)
+--R
+--R n + 1
+--R (a x + b)
+--R (2) --------------
+--R a n + a
+--R Type: Expression
Integer
+--E
+--S 92
+cc:=aa-bb
+--R
+--R n log(a x + b) n + 1
+--R (a x + b)%e - (a x + b)
+--R (3) ------------------------------------------
+--R a n + a
+--R Type: Expression
Integer
+--E
+@
+This messy formula can be simplified using the explog rule:
<<*>>=
---S 88
+--S 93
explog:=rule(%e^(n*log(x)) == x^n)
--R
--R n log(x) n
---R (2) %e == x
+--R (4) %e == x
--R Type: RewriteRule(Integer,Integer,Expression
Integer)
---E 88
-@
+--E
-<<*>>=
---S 89
-explog aa
+--S 94 14:80 Schaums and Axiom agree
+dd:=explog cc
--R
---R n
---R (a x + b)(a x + b)
---R (3) -------------------
---R a n + a
+--R n + 1 n
+--R - (a x + b) + (a x + b)(a x + b)
+--R (5) --------------------------------------
+--R a n + a
--R Type: Expression
Integer
---E 89
+--E
@
+The numerator is clearly zero but I cannot get Axiom to simplify it.
-\section{\cite{1}:14.81~~~~~$\displaystyle\int{x(ax+b)^n~dx}$}
-$$\int{x(ax+b)^n~dx}=
+\section{\cite{1}:14.81~~~~~$\displaystyle
+\int{x(ax+b)^n~dx}$}
+$$\int{x(ax+b)^n}=
\frac{(ax+b)^{n+2}}{(n+2)a^2}-\frac{b(ax+b)^{n+1}}{(n+1)a^2}
-{\rm\ provided\ }n \ne -1,-2$$
+{\rm\ provided\ }n \ne -1,-2
+$$
+<<*>>=
+)clear all
+--S 95
+aa:=integrate(x*(a*x+b)^n,x)
+--R
+--R 2 2 2 2 n log(a x + b)
+--R ((a n + a )x + a b n x - b )%e
+--R (1) ---------------------------------------------
+--R 2 2 2 2
+--R a n + 3a n + 2a
+--R Type: Union(Expression
Integer,...)
+--E
+
+--S 96
+bb:=((a*x+b)^(n+2))/((n+2)*a^2)-(b*(a*x+b)^(n+1))/((n+1)*a^2)
+--R
+--R n + 2 n + 1
+--R (n + 1)(a x + b) + (- b n - 2b)(a x + b)
+--R (2) --------------------------------------------------
+--R 2 2 2 2
+--R a n + 3a n + 2a
+--R Type: Expression
Integer
+--E
-\section{\cite{1}:14.82~~~~~$\displaystyle\int{x^2(ax+b)^n~dx}$}
-$$\int{x^2(ax+b)^n~dx}=
+--S 97
+cc:=aa-bb
+--R
+--R (3)
+--R 2 2 2 2 n log(a x + b) n
+ 2
+--R ((a n + a )x + a b n x - b )%e + (- n - 1)(a x + b)
+--R +
+--R n + 1
+--R (b n + 2b)(a x + b)
+--R /
+--R 2 2 2 2
+--R a n + 3a n + 2a
+--R Type: Expression
Integer
+--E
+
+--S 98
+explog:=rule(%e^(n*log(x)) == x^n)
+--R
+--R n log(x) n
+--R (4) %e == x
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 99 14:81 Schaums and Axiom agreement cannot be determined
+dd:=explog cc
+--R
+--R (5)
+--R n + 2 n + 1
+--R (- n - 1)(a x + b) + (b n + 2b)(a x + b)
+--R +
+--R 2 2 2 2 n
+--R ((a n + a )x + a b n x - b )(a x + b)
+--R /
+--R 2 2 2 2
+--R a n + 3a n + 2a
+--R Type: Expression
Integer
+--E
+@
+\section{\cite{1}:14.82~~~~~$\displaystyle
+\int{x^2(ax+b)^n~dx}$}
+$$\int{x^2(ax+b)^n}=
\frac{(ax+b)^{n+2}}{(n+3)a^3}-
\frac{2b(ax+b)^{n+2}}{(n+2)a^3}+
\frac{b^2(ax+b)^{n+1}}{(n+1)a^3}
-{\rm\ provided\ }n \ne -1,-2,-3$$
+{\rm\ provided\ }n \ne -1,-2,-3
+$$
<<*>>=
+)clear all
+--S 100
+aa:=integrate(x^2*(a*x+b)^n,x)
+--R
+--R (1)
+--R 3 2 3 3 3 2 2 2 2 2 3 n log(a x
+ b)
+--R ((a n + 3a n + 2a )x + (a b n + a b n)x - 2a b n x + 2b )%e
+--R
-----------------------------------------------------------------------------
+--R 3 3 3 2 3 3
+--R a n + 6a n + 11a n + 6a
+--R Type: Union(Expression
Integer,...)
+--E
+
+--S 101
+bb:=(a*x+b)^(n+3)/((n+3)*a^3)-(2*b*(a*x+b)^(n+2))/((n+2)*a^3)+(b^2*(a*x+b)^(n+1))/((n+1)*a^3)
+--R
+--R (2)
+--R 2 n + 3 2 n + 2
+--R (n + 3n + 2)(a x + b) + (- 2b n - 8b n - 6b)(a x + b)
+--R +
+--R 2 2 2 2 n + 1
+--R (b n + 5b n + 6b )(a x + b)
+--R /
+--R 3 3 3 2 3 3
+--R a n + 6a n + 11a n + 6a
+--R Type: Expression
Integer
+--E
+
+--S 102 14:82 Schaums and Axiom agreement cannot be determined
+cc:=aa-bb
+--R
+--R (3)
+--R 3 2 3 3 3 2 2 2 2 2 3
+--R ((a n + 3a n + 2a )x + (a b n + a b n)x - 2a b n x + 2b )
+--R *
+--R n log(a x + b)
+--R %e
+--R +
+--R 2 n + 3 2 n + 2
+--R (- n - 3n - 2)(a x + b) + (2b n + 8b n + 6b)(a x + b)
+--R +
+--R 2 2 2 2 n + 1
+--R (- b n - 5b n - 6b )(a x + b)
+--R /
+--R 3 3 3 2 3 3
+--R a n + 6a n + 11a n + 6a
+--R Type: Expression
Integer
+--E
+@
+\section{\cite{1}:14.83~~~~~$\displaystyle
+\int{x^m(ax+b)^n}~dx$}
+$$\int{x^m(ax+b)^n}
+\left\{
+\begin{array}{l}
+\displaystyle
+\frac{x^{m+1}(ax+b)^n}{m+n+1}
++\frac{nb}{m+n+1}\int{x^m(ax+b)^{n-1}}\\
+\\
+\displaystyle
+\frac{x^{m+1}(ax+b)^{n+1}}{(m+n+1)a}
+-\frac{mb}{(m+n+1)a}\int{x^{m-1}(ax+b)^n}\\
+\\
+\displaystyle
+\frac{-x^{m+1}(ax+b)^{n+1}}{(n+1)b}
++\frac{m+n+2}{(n+1)b}\int{x^m(ax+b)^{n+1}}\\
+\end{array}
+\right.
+$$
+
+<<*>>=
+--S 103 14:83 Axiom cannot do this integration
+aa:=integrate(x^m*(a*x+b)^n,x)
+--R
+--R x
+--R ++ m n
+--I (1) | %U (b + %U a) d%U
+--R ++
+--R Type: Union(Expression
Integer,...)
+--E
+
)spool
)lisp (bye)
@
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