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}