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) @