diff --git a/changelog b/changelog index f2d93ae..c467da5 100644 --- a/changelog +++ b/changelog @@ -1,3 +1,9 @@ +20080328 tpd src/input/Makefile add integration regression testing +20080328 tpd src/input/schaum6.input integrals of x^2+a^2 +20080328 tpd src/input/schaum5.input integrals of sqrt(ax+b) and sqrt(px+q) +20080328 tpd src/input/schaum4.input integrals of sqrt(ax+b) and px+q +20080328 tpd src/input/schaum3.input integrals of ax+b and px+q +20080328 tpd src/input/schaum2.input integrals of sqrt(ax+b) 20080325 tpd Makefile VERSION="Axiom (March 2008)" 20080325 tpd src/algebra/axserver.spad set up handling of operations pages 20080325 tpd src/interp/interp-proclaims.lisp case-change display diff --git a/src/input/Makefile.pamphlet b/src/input/Makefile.pamphlet index 2b626c3..d4caed2 100644 --- a/src/input/Makefile.pamphlet +++ b/src/input/Makefile.pamphlet @@ -355,7 +355,8 @@ 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 \ + schaum1.regress schaum2.regress schaum3.regress schaum4.regress \ + schaum5.regress schaum6.regress \ scherk.regress scope.regress seccsc.regress \ segbind.regress seg.regress \ series2.regress series.regress sersolve.regress set.regress \ @@ -628,6 +629,8 @@ FILES= ${OUT}/algaggr.input ${OUT}/algbrbf.input ${OUT}/algfacob.input \ ${OUT}/reclos.input ${OUT}/regset.input \ ${OUT}/robidoux.input ${OUT}/roman.input ${OUT}/roots.input \ ${OUT}/ruleset.input ${OUT}/rules.input ${OUT}/schaum1.input \ + ${OUT}/schaum2.input ${OUT}/schaum3.input ${OUT}/schaum4.input \ + ${OUT}/schaum5.input ${OUT}/schaum6.input \ ${OUT}/saddle.input \ ${OUT}/scherk.input ${OUT}/scope.input ${OUT}/seccsc.input \ ${OUT}/segbind.input ${OUT}/seg.input ${OUT}/series2.input \ @@ -926,7 +929,9 @@ 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}/schaum1.input.dvi ${DOC}/schaum2.input.dvi \ + ${DOC}/schaum3.input.dvi ${DOC}/schaum4.input.dvi \ + ${DOC}/schaum5.input.dvi ${DOC}/schaum6.input.dvi \ ${DOC}/s01eaf.input.dvi ${DOC}/s13aaf.input.dvi \ ${DOC}/s13acf.input.dvi ${DOC}/s13adf.input.dvi \ ${DOC}/s14aaf.input.dvi ${DOC}/s14abf.input.dvi \ diff --git a/src/input/schaum2.input.pamphlet b/src/input/schaum2.input.pamphlet new file mode 100644 index 0000000..cb8e6db --- /dev/null +++ b/src/input/schaum2.input.pamphlet @@ -0,0 +1,1464 @@ +\documentclass{article} +\usepackage{axiom} +\begin{document} +\title{\$SPAD/input schaum2.input} +\author{Timothy Daly} +\maketitle +\eject +\tableofcontents +\eject +\section{\cite{1}:14.84~~~~~$\displaystyle\int{\frac{dx}{\sqrt{ax+b}}}$} +$$\int{\frac{dx}{\sqrt{ax+b}}}=\frac{2\sqrt{ax+b}}{a}$$ +<<*>>= +)spool schaum2.output +)set message test on +)set message auto off +)clear all + +--S 1 of 92 +aa:=integrate(1/sqrt(a*x+b),x) +--R +--R +--R +-------+ +--R 2\|a x + b +--R (1) ----------- +--R a +--R Type: Union(Expression Integer,...) +--E +@ +<<*>>= +--S 2 of 92 +bb:=(2*sqrt(a*x+b))/a +--R +--R +--R +-------+ +--R 2\|a x + b +--R (2) ----------- +--R a +--R Type: Expression Integer +--E +@ +<<*>>= +--S 3 of 92 +cc:=aa-bb +--R +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.85~~~~~$\displaystyle\int{\frac{x~dx}{\sqrt{ax+b}}}$} +$$\int{\frac{x~dx}{\sqrt{ax+b}}}=\frac{2(ax-2b)}{3a^2}\sqrt{ax+b}$$ +<<*>>= +)clear all + +--S 4 of 92 +aa:=integrate(x/sqrt(a*x+b),x) +--R +--R +--R +-------+ +--R (2a x - 4b)\|a x + b +--R (1) --------------------- +--R 2 +--R 3a +--R Type: Union(Expression Integer,...) +--E +@ +<<*>>= +--S 5 of 92 +bb:=(2*(a*x-2*b))/(3*a^2)*sqrt(a*x+b) +--R +--R +--R +-------+ +--R (2a x - 4b)\|a x + b +--R (2) --------------------- +--R 2 +--R 3a +--R Type: Expression Integer +--E +@ +<<*>>= +--S 6 of 92 +cc:=aa-bb +--R +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.86~~~~~$\displaystyle\int{\frac{x^2~dx}{\sqrt{ax+b}}}$} +$$\int{\frac{x~dx}{\sqrt{ax+b}}}= +\frac{2(3a^2x^2-4abx+8b^2)}{15a^2}\sqrt{ax+b}$$ +<<*>>= +)clear all + +--S 7 of 92 +aa:=integrate(x^2/sqrt(a*x+b),x) +--R +--R +--R 2 2 2 +-------+ +--R (6a x - 8a b x + 16b )\|a x + b +--R (1) --------------------------------- +--R 3 +--R 15a +--R Type: Union(Expression Integer,...) +--E +@ +<<*>>= +--S 8 of 92 +bb:=(2*(3*a^2*x^2-4*a*b*x+8*b^2))/(15*a^3)*sqrt(a*x+b) +--R +--R +--R 2 2 2 +-------+ +--R (6a x - 8a b x + 16b )\|a x + b +--R (2) --------------------------------- +--R 3 +--R 15a +--R Type: Expression Integer +--E +@ +<<*>>= +--S 9 of 92 +cc:=aa-bb +--R +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.87~~~~~$\displaystyle\int{\frac{dx}{x\sqrt{ax+b}}}$} +$$\int{\frac{dx}{x\sqrt{ax+b}}}= +\left\{ +\begin{array}{l} +\displaystyle +\frac{1}{\sqrt{b}}~\ln +\left(\frac{\sqrt{ax+b}-\sqrt{b}}{\sqrt{ax+b}+\sqrt{b}}\right)\\ +\displaystyle +\frac{2}{\sqrt{-b}}~\tan^{-1}\sqrt{\frac{ax+b}{-b}} +\end{array} +\right.$$ + +Note: the first answer assumes $b > 0$ and the second assumes $b < 0$. +<<*>>= +)clear all + +--S 10 of 92 +aa:=integrate(1/(x*sqrt(a*x+b)),x) +--R +--R +--R +-------+ +-+ +---+ +-------+ +--R - 2b\|a x + b + (a x + 2b)\|b \|- b \|a x + b +--R log(-------------------------------) 2atan(----------------) +--R x b +--R (1) [------------------------------------,- -----------------------] +--R +-+ +---+ +--R \|b \|- b +--R Type: Union(List Expression Integer,...) +--E +@ +Cleary Spiegel's first answer assumes $b > 0$: +<<*>>= +--S 11 of 92 +bb1:=1/sqrt(b)*log((sqrt(a*x+b)-sqrt(b))/(sqrt(a*x+b)+sqrt(b))) +--R +--R +--R +-------+ +-+ +--R \|a x + b - \|b +--R log(-----------------) +--R +-------+ +-+ +--R \|a x + b + \|b +--R (2) ---------------------- +--R +-+ +--R \|b +--R Type: Expression Integer +--E +@ +So we try the difference of the two results +<<*>>= +--S 12 of 92 +cc11:=aa.1-bb1 +--R +--R +-------+ +-+ +-------+ +-+ +--R \|a x + b - \|b - 2b\|a x + b + (a x + 2b)\|b +--R - log(-----------------) + log(-------------------------------) +--R +-------+ +-+ x +--R \|a x + b + \|b +--R (3) --------------------------------------------------------------- +--R +-+ +--R \|b +--R Type: Expression Integer +--E +@ +But the results don't simplify to 0. So we try some other tricks. + +Since both functions are of the form log(f(x))/sqrt(b) we extract +the f(x) from each. First we get the function from Axiom's first answer: +<<*>>= +--S 13 of 92 +ff:=exp(aa.1*sqrt(b)) +--R +--R +-------+ +-+ +--R - 2b\|a x + b + (a x + 2b)\|b +--R (4) ------------------------------- +--R x +--R Type: Expression Integer +--E +@ +and we get the same form from Spiegel's answer +<<*>>= +--S 14 of 92 +gg:=exp(bb1*sqrt(b)) +--R +--R +-------+ +-+ +--R \|a x + b - \|b +--R (5) ----------------- +--R +-------+ +-+ +--R \|a x + b + \|b +--R Type: Expression Integer +--E +@ +We can change Spiegel's form into Axiom's form because they differ by +the constant a*sqrt(b). To see this we multiply the numerator and +denominator by $1 == (sqrt(a*x+b) - sqrt(b))/(sqrt(a*x+b) - sqrt(b))$. + +First we multiply the numerator by $(sqrt(a*x+b) - sqrt(b))$ +<<*>>= +--S 15 of 92 +gg1:=gg*(sqrt(a*x+b) - sqrt(b)) +--R +--R +-+ +-------+ +--R - 2\|b \|a x + b + a x + 2b +--R (6) ---------------------------- +--R +-------+ +-+ +--R \|a x + b + \|b +--R Type: Expression Integer +--E +@ +Now we multiply the denominator by $(sqrt(a*x+b) - sqrt(b))$ +<<*>>= +--S 16 of 92 +gg2:=gg1/(sqrt(a*x+b) - sqrt(b)) +--R +--R +-+ +-------+ +--R - 2\|b \|a x + b + a x + 2b +--R (7) ---------------------------- +--R a x +--R Type: Expression Integer +--E +@ +and now we multiply by the integration constant $a*sqrt(b)$ +<<*>>= +--S 17 of 92 +gg3:=gg2*(a*sqrt(b)) +--R +--R +-------+ +-+ +--R - 2b\|a x + b + (a x + 2b)\|b +--R (8) ------------------------------- +--R x +--R Type: Expression Integer +--E +@ +and when we difference this with ff, the Axiom answer we get: +<<*>>= +--S 18 of 92 +ff-gg3 +--R +--R (9) 0 +--R Type: Expression Integer +--E +@ +So the constant of integration difference is $a*sqrt(b)$ + +Now we look at the second equations. We difference Axiom's second answer +from Spiegel's answer: +<<*>>= +--S 19 of 92 +t1:=aa.2-bb1 +--R +--R +-------+ +-+ +---+ +-------+ +--R +---+ \|a x + b - \|b +-+ \|- b \|a x + b +--R - \|- b log(-----------------) - 2\|b atan(----------------) +--R +-------+ +-+ b +--R \|a x + b + \|b +--R (10) ------------------------------------------------------------ +--R +---+ +-+ +--R \|- b \|b +--R Type: Expression Integer +--E +@ +and again they do not simplify to zero. But we can show that both answers +differ by a constant because the derivative is zero: +<<*>>= +--S 20 of 92 +D(t1,x) +--R +--R (11) 0 +--R Type: Expression Integer +--E +@ + +Rather than find the constant this time we will differentiate both +answers and compare them with the original equation. +<<*>>= +--S 21 of 92 +target:=1/(x*sqrt(a*x+b)) +--R +--R 1 +--R (12) ----------- +--R +-------+ +--R x\|a x + b +--R Type: Expression Integer +--E +@ +and we select the second Axiom solution +<<*>>= +--S 22 of 92 +aa2:=aa.2 +--R +--R +---+ +-------+ +--R \|- b \|a x + b +--R 2atan(----------------) +--R b +--R (13) - ----------------------- +--R +---+ +--R \|- b +--R Type: Expression Integer +--E +@ +take its derivative +<<*>>= +--S 23 of 92 +ad2:=D(aa2,x) +--R +--R 1 +--R (14) ----------- +--R +-------+ +--R x\|a x + b +--R Type: Expression Integer +--E +@ +When we take the difference of Axiom's input and the derivative of the +output we see: +<<*>>= +--S 24 of 92 +ad2-target +--R +--R (15) 0 +--R Type: Expression Integer +--E +@ +Thus the original equation and Axiom's derivative of the integral are equal. + +Now we do the same with Spiegel's answer. We take the derivative of his +answer. +<<*>>= +--S 25 of 92 +ab1:=D(bb1,x) +--R +--R +-------+ +-+ +--R \|a x + b + \|b +--R (16) ---------------------------- +--R +-+ +-------+ 2 +--R x\|b \|a x + b + a x + b x +--R Type: Expression Integer +--E +@ +and we difference it from the original equation +<<*>>= +--S 26 of 92 +ab1-target +--R +--R (17) 0 +--R Type: Expression Integer +--E +@ +Thus the original equation and Spiegel's derivative of the integral are equal. + +So we can conclude that both second answers are correct although they differ +by a constant of integration. + + \section{\cite{1}:14.88~~~~~$\displaystyle\int{\frac{dx}{x^2\sqrt{ax+b}}}$} +$$\int{\frac{dx}{x^2\sqrt{ax+b}}}= +-\frac{\sqrt{ax+b}}{bx}-\frac{a}{2b}~\int{\frac{dx}{x\sqrt{ax+b}}}$$ +<<*>>= +)clear all + +--S 27 of 92 +aa:=integrate(1/(x^2*sqrt(a*x+b)),x) +--R +--R +--R (1) +--R +-------+ +-+ +--R 2b\|a x + b + (a x + 2b)\|b +-+ +-------+ +--R a x log(-----------------------------) - 2\|b \|a x + b +--R x +--R [--------------------------------------------------------, +--R +-+ +--R 2b x\|b +--R +---+ +-------+ +--R \|- b \|a x + b +---+ +-------+ +--R a x atan(----------------) - \|- b \|a x + b +--R b +--R ---------------------------------------------] +--R +---+ +--R b x\|- b +--R Type: Union(List Expression Integer,...) +--E +@ + +In order to write down the book answer we need to first take the +integral which has two results +<<*>>= +--S 28 of 92 +dd:=integrate(1/(x*sqrt(a*x+b)),x) +--R +--R +--R +-------+ +-+ +---+ +-------+ +--R - 2b\|a x + b + (a x + 2b)\|b \|- b \|a x + b +--R log(-------------------------------) 2atan(----------------) +--R x b +--R (2) [------------------------------------,- -----------------------] +--R +-+ +---+ +--R \|b \|- b +--R Type: Union(List Expression Integer,...) +--E +@ +and derive two results for the book answer. The first result assumes +$b > 0$ +<<*>>= +--S 29 of 92 +bb1:=-sqrt(a*x+b)/(b*x)-a/(2*b)*dd.1 +--R +--R +--R +-------+ +-+ +--R - 2b\|a x + b + (a x + 2b)\|b +-+ +-------+ +--R - a x log(-------------------------------) - 2\|b \|a x + b +--R x +--R (3) ------------------------------------------------------------ +--R +-+ +--R 2b x\|b +--R Type: Expression Integer +--E +@ +and the second result assumes $b < 0$. +<<*>>= +--S 30 of 92 +bb2:=-sqrt(a*x+b)/(b*x)-a/(2*b)*dd.2 +--R +--R +--R +---+ +-------+ +--R \|- b \|a x + b +---+ +-------+ +--R a x atan(----------------) - \|- b \|a x + b +--R b +--R (4) --------------------------------------------- +--R +---+ +--R b x\|- b +--R Type: Expression Integer +--E +@ + +So we compute the difference of Axiom's first result with Spiegel's +first result +<<*>>= +--S 31 of 92 +cc11:=bb1-aa.1 +--R +--R (5) +--R +-------+ +-+ +--R 2b\|a x + b + (a x + 2b)\|b +--R - a log(-----------------------------) +--R x +--R + +--R +-------+ +-+ +--R - 2b\|a x + b + (a x + 2b)\|b +--R - a log(-------------------------------) +--R x +--R / +--R +-+ +--R 2b\|b +--R Type: Expression Integer +--E +@ +we compute its derivative +<<*>>= +--S 32 of 92 +D(cc11,x) +--R +--R (6) 0 +--R Type: Expression Integer +--E +@ +and we can see that the answers differ by a constant, the constant of +integration. So Axiom's first answer should differentiate back to the target +equation. +<<*>>= +--S 33 of 92 +target:=1/(x^2*sqrt(a*x+b)) +--R +--R 1 +--R (7) ------------ +--R 2 +-------+ +--R x \|a x + b +--R Type: Expression Integer +--E +@ +We differentiate Axiom's first answer +<<*>>= +--S 34 of 92 +ad1:=D(aa.1,x) +--R +--R +-+ +-------+ 2 +--R (a x + 2b)\|b \|a x + b + 2a b x + 2b +--R (8) ---------------------------------------------------------- +--R 3 2 2 +-------+ 2 4 3 2 2 +-+ +--R (2a b x + 2b x )\|a x + b + (a x + 3a b x + 2b x )\|b +--R Type: Expression Integer +--E +@ +and subtract it from the target equation +<<*>>= +--S 35 of 92 +ad1-target +--R +--R (9) 0 +--R Type: Expression Integer +--E +@ +and now we do the same with first Spiegel's answer: +<<*>>= +--S 36 of 92 +bd1:=D(bb1,x) +--R +--R +-+ +-------+ 2 +--R (- a x - 2b)\|b \|a x + b + 2a b x + 2b +--R (10) ------------------------------------------------------------ +--R 3 2 2 +-------+ 2 4 3 2 2 +-+ +--R (2a b x + 2b x )\|a x + b + (- a x - 3a b x - 2b x )\|b +--R Type: Expression Integer +--E +@ +and we subtract it from the target +<<*>>= +--S 37 of 92 +bd1-target +--R +--R (11) 0 +--R Type: Expression Integer +--E +@ +so we know that the two first answers are both correct and that their +integrals differ by a constant. + +Now we look at the second answers. We difference the answers and can +see immediately that they are equal. +<<*>>= +--S 38 of 92 +cc22:=bb2-aa.2 +--R +--R +--R (12) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.89~~~~~$\displaystyle\int{\sqrt{ax+b}~dx}$} +$$\int{\sqrt{ax+b}~dx}= +\frac{2\sqrt{(ax+b)^3}}{3a}$$ +<<*>>= +)clear all + +--S 39 of 92 +aa:=integrate(sqrt(a*x+b),x) +--R +--R +--R +-------+ +--R (2a x + 2b)\|a x + b +--R (1) --------------------- +--R 3a +--R Type: Union(Expression Integer,...) +--E +@ +<<*>>= +--S 40 of 92 +bb:=(2*sqrt((a*x+b)^3))/(3*a) +--R +--R +--R +----------------------------+ +--R | 3 3 2 2 2 3 +--R 2\|a x + 3a b x + 3a b x + b +--R (2) -------------------------------- +--R 3a +--R Type: Expression Integer +--E +@ +<<*>>= +--S 41 of 92 +cc:=aa-bb +--R +--R +----------------------------+ +--R | 3 3 2 2 2 3 +-------+ +--R - 2\|a x + 3a b x + 3a b x + b + (2a x + 2b)\|a x + b +--R (3) ---------------------------------------------------------- +--R 3a +--R Type: Expression Integer +--E +@ +Since this didn't simplify we could check each answer using the derivative +<<*>>= +--S 42 of 92 +target:=sqrt(a*x+b) +--R +--R +-------+ +--R (4) \|a x + b +--R Type: Expression Integer +--E +@ +We take the derivative of Axiom's answer +<<*>>= +--S 43 of 92 +t1:=D(aa,x) +--R +--R a x + b +--R (5) ---------- +--R +-------+ +--R \|a x + b +--R Type: Expression Integer +--E +@ +And we subtract the target from the derivative of Axiom's answer +<<*>>= +--S 44 of 92 +t1-target +--R +--R (6) 0 +--R Type: Expression Integer +--E +@ +So they are equal. Now we do the same with Spiegel's answer +<<*>>= +--S 45 of 92 +t2:=D(bb,x) +--R +--R 2 2 2 +--R a x + 2a b x + b +--R (7) ------------------------------- +--R +----------------------------+ +--R | 3 3 2 2 2 3 +--R \|a x + 3a b x + 3a b x + b +--R Type: Expression Integer +--E +@ +The numerator is +<<*>>= +--S 46 of 92 +nn:=(a*x+b)^2 +--R +--R 2 2 2 +--R (8) a x + 2a b x + b +--R Type: Polynomial Integer +--E +@ +<<*>>= +--S 47 of 92 +mm:=(a*x+b)^3 +--R +--R 3 3 2 2 2 3 +--R (9) a x + 3a b x + 3a b x + b +--R Type: Polynomial Integer +--E +@ +which expands to Spiegel's version. +<<*>>= +--S 48 of 92 +result=nn/sqrt(mm) +--R +--R 2 2 2 +--R a x + 2a b x + b +--R (10) result= ------------------------------- +--R +----------------------------+ +--R | 3 3 2 2 2 3 +--R \|a x + 3a b x + 3a b x + b +--R Type: Equation Expression Integer +--E +@ +and this reduces to $\sqrt{ax+b}$ + +\section{\cite{1}:14.90~~~~~$\displaystyle\int{x\sqrt{ax+b}~dx}$} +$$\int{x\sqrt{ax+b}~dx}= +\frac{2(3ax-2b)}{15a^2}~\sqrt{(ax+b)^3}$$ +<<*>>= +)clear all + +--S 49 of 92 +aa:=integrate(x*sqrt(a*x+b),x) +--R +--R +--R 2 2 2 +-------+ +--R (6a x + 2a b x - 4b )\|a x + b +--R (1) -------------------------------- +--R 2 +--R 15a +--R Type: Union(Expression Integer,...) +--E +@ +<<*>>= +--S 50 of 92 +bb:=(2*(3*a*x-2*b))/(15*a^2)*sqrt((a*x+b)^3) +--R +--R +--R +----------------------------+ +--R | 3 3 2 2 2 3 +--R (6a x - 4b)\|a x + 3a b x + 3a b x + b +--R (2) ------------------------------------------ +--R 2 +--R 15a +--R Type: Expression Integer +--E +@ +<<*>>= +--S 51 of 92 +cc:=aa-bb +--R +--R (3) +--R +----------------------------+ +--R | 3 3 2 2 2 3 +--R (- 6a x + 4b)\|a x + 3a b x + 3a b x + b +--R + +--R 2 2 2 +-------+ +--R (6a x + 2a b x - 4b )\|a x + b +--R / +--R 2 +--R 15a +--R Type: Expression Integer +--E +@ +If we had the terms +<<*>>= +--S 52 of 92 +t1:=(3*a*x-2*b) +--R +--R (4) 3a x - 2b +--R Type: Polynomial Integer +--E +@ +<<*>>= +--S 53 of 92 +t2:=(a*x+b) +--R +--R (5) a x + b +--R Type: Polynomial Integer +--E +@ +We can construct the Axiom result +<<*>>= +--S 54 of 92 +2*t1*t2*sqrt(t2)/(15*a^2) +--R +--R 2 2 2 +-------+ +--R (6a x + 2a b x - 4b )\|a x + b +--R (6) -------------------------------- +--R 2 +--R 15a +--R Type: Expression Integer +--E +@ +and we can construct the Spiegel result +<<*>>= +--S 55 of 92 +2*t1*sqrt(t2^3)/(15*a^2) +--R +--R +----------------------------+ +--R | 3 3 2 2 2 3 +--R (6a x - 4b)\|a x + 3a b x + 3a b x + b +--R (7) ------------------------------------------ +--R 2 +--R 15a +--R Type: Expression Integer +--E +@ +the difference of these two depends on +<<*>>= +--S 56 of 92 +t2*sqrt(t2)-sqrt(t2^3) +--R +--R +----------------------------+ +--R | 3 3 2 2 2 3 +-------+ +--R (8) - \|a x + 3a b x + 3a b x + b + (a x + b)\|a x + b +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.91~~~~~$\displaystyle\int{x^2\sqrt{ax+b}~dx}$} +$$\int{x^2\sqrt{ax+b}~dx}= +\frac{2(15a^2x^2-12abx+8b^2)}{105a^2}~\sqrt{(a+bx)^3}$$ +Note: the sqrt term is almost certainly $\sqrt{(ax+b)}$ +<<*>>= +)clear all + +--S 57 of 92 +aa:=integrate(x^2*sqrt(a*x+b),x) +--R +--R +--R 3 3 2 2 2 3 +-------+ +--R (30a x + 6a b x - 8a b x + 16b )\|a x + b +--R (1) -------------------------------------------- +--R 3 +--R 105a +--R Type: Union(Expression Integer,...) +--E +@ +<<*>>= +--S 58 of 92 +bb:=(2*(15*a^2*x^2-12*a*b*x+8*b^2))/(105*a^2)*sqrt((a*x+b)^3) +--R +--R +--R +----------------------------+ +--R 2 2 2 | 3 3 2 2 2 3 +--R (30a x - 24a b x + 16b )\|a x + 3a b x + 3a b x + b +--R (2) -------------------------------------------------------- +--R 2 +--R 105a +--R Type: Expression Integer +--E +@ +<<*>>= +--S 59 of 92 +cc:=aa-bb +--R +--R +--R (3) +--R +----------------------------+ +--R 3 2 2 2 | 3 3 2 2 2 3 +--R (- 30a x + 24a b x - 16a b )\|a x + 3a b x + 3a b x + b +--R + +--R 3 3 2 2 2 3 +-------+ +--R (30a x + 6a b x - 8a b x + 16b )\|a x + b +--R / +--R 3 +--R 105a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.92~~~~~$\displaystyle\int{\frac{\sqrt{ax+b}}{x}~dx}$} +$$\int{\frac{\sqrt{ax+b}}{x}~dx}= +2\sqrt{ax+b}+b~\int{\frac{dx}{x\sqrt{ax+b}}}$$ +<<*>>= +)clear all + +--S 60 of 92 +aa:=integrate(sqrt(a*x+b)/x,x) +--R +--R +--R (1) +--R +-+ +-------+ +--R +-+ - 2\|b \|a x + b + a x + 2b +-------+ +--R [\|b log(----------------------------) + 2\|a x + b , +--R x +--R +-------+ +--R +---+ \|a x + b +-------+ +--R - 2\|- b atan(----------) + 2\|a x + b ] +--R +---+ +--R \|- b +--R Type: Union(List Expression Integer,...) +--E +@ +<<*>>= +--S 61 of 92 +dd:=integrate(1/(x*sqrt(a*x+b)),x) +--R +--R +--R +-------+ +-+ +---+ +-------+ +--R - 2b\|a x + b + (a x + 2b)\|b \|- b \|a x + b +--R log(-------------------------------) 2atan(----------------) +--R x b +--R (2) [------------------------------------,- -----------------------] +--R +-+ +---+ +--R \|b \|- b +--R Type: Union(List Expression Integer,...) +--E +@ +<<*>>= +--S 62 of 92 +bb1:=2*sqrt(a*x+b)+b*dd.1 +--R +--R +--R +-------+ +-+ +--R - 2b\|a x + b + (a x + 2b)\|b +-+ +-------+ +--R b log(-------------------------------) + 2\|b \|a x + b +--R x +--R (3) -------------------------------------------------------- +--R +-+ +--R \|b +--R Type: Expression Integer +--E +@ +<<*>>= +--S 63 of 92 +bb2:=2*sqrt(a*x+b)+b*dd.2 +--R +--R +--R +---+ +-------+ +--R \|- b \|a x + b +---+ +-------+ +--R - 2b atan(----------------) + 2\|- b \|a x + b +--R b +--R (4) ----------------------------------------------- +--R +---+ +--R \|- b +--R Type: Expression Integer +--E +@ +<<*>>= +--S 64 of 92 +cc11:=bb1-aa.1 +--R +--R +--R (5) +--R +-------+ +-+ +-+ +-------+ +--R - 2b\|a x + b + (a x + 2b)\|b - 2\|b \|a x + b + a x + 2b +--R b log(-------------------------------) - b log(----------------------------) +--R x x +--R ---------------------------------------------------------------------------- +--R +-+ +--R \|b +--R Type: Expression Integer +--E +@ +<<*>>= +--S 65 of 92 +cc12:=bb1-aa.2 +--R +--R +--R +-------+ +-+ +-------+ +--R - 2b\|a x + b + (a x + 2b)\|b +---+ +-+ \|a x + b +--R b log(-------------------------------) + 2\|- b \|b atan(----------) +--R x +---+ +--R \|- b +--R (6) -------------------------------------------------------------------- +--R +-+ +--R \|b +--R Type: Expression Integer +--E +@ +<<*>>= +--S 66 of 92 +cc21:=bb2-aa.1 +--R +--R +--R (7) +--R +-+ +-------+ +---+ +-------+ +--R +---+ +-+ - 2\|b \|a x + b + a x + 2b \|- b \|a x + b +--R - \|- b \|b log(----------------------------) - 2b atan(----------------) +--R x b +--R ------------------------------------------------------------------------- +--R +---+ +--R \|- b +--R Type: Expression Integer +--E +@ +<<*>>= +--S 67 of 92 +cc22:=bb2-aa.2 +--R +--R +--R +---+ +-------+ +-------+ +--R \|- b \|a x + b \|a x + b +--R - 2b atan(----------------) - 2b atan(----------) +--R b +---+ +--R \|- b +--R (8) ------------------------------------------------- +--R +---+ +--R \|- b +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.93~~~~~$\displaystyle\int{\frac{\sqrt{ax+b}}{x^2}~dx}$} +$$\int{\frac{\sqrt{ax+b}}{x^2}~dx}= +-\frac{\sqrt{ax+b}}{x}+\frac{a}{2}~\int{\frac{dx}{x\sqrt{ax+b}}}$$ +<<*>>= +)clear all + +--S 68 of 92 +aa:=integrate(sqrt(a*x+b)/x^2,x) +--R +--R +--R (1) +--R +-------+ +-+ +--R - 2b\|a x + b + (a x + 2b)\|b +-+ +-------+ +--R a x log(-------------------------------) - 2\|b \|a x + b +--R x +--R [----------------------------------------------------------, +--R +-+ +--R 2x\|b +--R +---+ +-------+ +--R \|- b \|a x + b +---+ +-------+ +--R - a x atan(----------------) - \|- b \|a x + b +--R b +--R -----------------------------------------------] +--R +---+ +--R x\|- b +--R Type: Union(List Expression Integer,...) +--E +@ +<<*>>= +--S 69 of 92 +dd:=integrate(1/(x*sqrt(a*x+b)),x) +--R +--R +--R +-------+ +-+ +---+ +-------+ +--R - 2b\|a x + b + (a x + 2b)\|b \|- b \|a x + b +--R log(-------------------------------) 2atan(----------------) +--R x b +--R (2) [------------------------------------,- -----------------------] +--R +-+ +---+ +--R \|b \|- b +--R Type: Union(List Expression Integer,...) +--E +@ +<<*>>= +--S 70 of 92 +bb1:=-sqrt(a*x+b)/x+a/2*dd.1 +--R +--R +--R +-------+ +-+ +--R - 2b\|a x + b + (a x + 2b)\|b +-+ +-------+ +--R a x log(-------------------------------) - 2\|b \|a x + b +--R x +--R (3) ---------------------------------------------------------- +--R +-+ +--R 2x\|b +--R Type: Expression Integer +--E +@ +<<*>>= +--S 71 of 92 +bb2:=-sqrt(a*x+b)/x+a/2*dd.2 +--R +--R +--R +---+ +-------+ +--R \|- b \|a x + b +---+ +-------+ +--R - a x atan(----------------) - \|- b \|a x + b +--R b +--R (4) ----------------------------------------------- +--R +---+ +--R x\|- b +--R Type: Expression Integer +--E +@ +<<*>>= +--S 72 of 92 +cc11:=bb1-aa.1 +--R +--R +--R (5) 0 +--R Type: Expression Integer +--E +@ +<<*>>= +--S 73 of 92 +cc21:=bb-aa.1 +--R +--R +--R (6) +--R +-------+ +-+ +--R - 2b\|a x + b + (a x + 2b)\|b +-+ +-------+ +-+ +--R - a x log(-------------------------------) + 2\|b \|a x + b + 2bb x\|b +--R x +--R ------------------------------------------------------------------------ +--R +-+ +--R 2x\|b +--R Type: Expression Integer +--E +@ +<<*>>= +--S 74 of 92 +cc12:=bb1-aa.2 +--R +--R +--R (7) +--R +-------+ +-+ +---+ +-------+ +--R +---+ - 2b\|a x + b + (a x + 2b)\|b +-+ \|- b \|a x + b +--R a\|- b log(-------------------------------) + 2a\|b atan(----------------) +--R x b +--R -------------------------------------------------------------------------- +--R +---+ +-+ +--R 2\|- b \|b +--R Type: Expression Integer +--E +@ +<<*>>= +--S 75 of 92 +cc22:=bb2-aa.2 +--R +--R +--R (8) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.94~~~~~$\displaystyle\int{\frac{x^m}{\sqrt{ax+b}}~dx}$} +$$\int{\frac{x^m}{\sqrt{ax+b}}~dx}= +\frac{2x^m\sqrt{ax+b}}{(2m+1)a}-\frac{2mb}{(2m+1)a} +~\int{\frac{x^{m-1}}{\sqrt{ax+b}}~dx}$$ +<<*>>= +)clear all + +--S 76 of 92 +aa:=integrate(x^m/sqrt(a*x+b),x) +--R +--R +--R x m +--I ++ %L +--I (1) | ----------- d%L +--R ++ +--------+ +--I \|b + %L a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.95~~~~~$\displaystyle\int{\frac{dx}{x^m\sqrt{ax+b}}}$} +$$\int{\frac{dx}{x^m\sqrt{ax+b}}}= +-\frac{\sqrt{ax+b}}{(m-1)bx^{m-1}}-\frac{(2m-3)a}{(2m-2)b} +~\int{\frac{dx}{x^{m-1}\sqrt{ax+b}}}$$ +<<*>>= +)clear all + +--S 77 of 92 +aa:=integrate(1/(x^m*sqrt(a*x+b)),x) +--R +--R +--R x +--R ++ 1 +--I (1) | -------------- d%L +--R ++ m +--------+ +--I %L \|b + %L a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.96~~~~~$\displaystyle\int{x^m\sqrt{ax+b}~dx}$} +$$\int{x^m\sqrt{ax+b}~dx}= +\frac{2x^m}{(2m+3)a}(ax+b)^{3/2} +-\frac{2mb}{(2m+3)a}~\int{x^{m-1}\sqrt{ax+b}~dx}$$ +<<*>>= +)clear all + +--S 78 of 92 +aa:=integrate(x^m*sqrt(a*x+b),x) +--R +--R +--R x +--R ++ m +--------+ +--I (1) | %L \|b + %L a d%L +--R ++ +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.97~~~~~$\displaystyle\int{\frac{\sqrt{ax+b}}{x^m}~dx}$} +$$\int{\frac{\sqrt{ax+b}}{x^m}~dx}= +-\frac{\sqrt{ax+b}}{(m-1)x^{m-1}} ++\frac{a}{2(m-1)}~\int{\frac{dx}{x^{m-1}\sqrt{ax+b}}}$$ +<<*>>= +)clear all + +--S 79 of 92 +aa:=integrate(sqrt(a*x+b)/x^m,x) +--R +--R +--R x +--------+ +--I ++ \|b + %L a +--I (1) | ----------- d%L +--R ++ m +--I %L +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.98~~~~~$\displaystyle\int{\frac{\sqrt{ax+b}}{x^m}~dx}$} +$$\int{\frac{\sqrt{ax+b}}{x^m}~dx}= +\frac{-(ax+b)^{3/2}}{(m-1)bx^{m-1}} +-\frac{(2m-5)a}{(2m-2)b}~\int{\frac{\sqrt{ax+b}}{x^{m-1}}~dx}$$ +Note: 14.98 is the same as 14.97 +<<*>>= +)clear all + +--S 80 of 92 +aa:=integrate(sqrt(a*x+b)/x^m,x) +--R +--R +--R x +--------+ +--I ++ \|b + %L a +--I (1) | ----------- d%L +--R ++ m +--I %L +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.99~~~~~$\displaystyle\int{(ax+b)^{m/2}~dx}$} +$$\int{(ax+b)^{m/2}~dx}= +\frac{2(ax+b)^{(m+2)/2}}{a(m+2)}$$ +<<*>>= +)clear all + +--S 81 of 92 +aa:=integrate((a*x+b)^(m/2),x) +--R +--R +--R m log(a x + b) +--R -------------- +--R 2 +--R (2a x + 2b)%e +--R (1) --------------------------- +--R a m + 2a +--R Type: Union(Expression Integer,...) +--E +@ +<<*>>= +--S 82 of 92 +bb:=(2*(a*x+b)^((m+2)/2))/(a*(m+2)) +--R +--R +--R m + 2 +--R ----- +--R 2 +--R 2(a x + b) +--R (2) --------------- +--R a m + 2a +--R Type: Expression Integer +--E +@ +<<*>>= +--S 83 of 92 +cc:=aa-bb +--R +--R +--R m log(a x + b) m + 2 +--R -------------- ----- +--R 2 2 +--R (2a x + 2b)%e - 2(a x + b) +--R (3) --------------------------------------------- +--R a m + 2a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.100~~~~~$\displaystyle\int{x(ax+b)^{m/2}~dx}$} +$$\int{x(ax+b)^{m/2}~dx}= +\frac{2(ax+b)^{(m+4)/2}}{a^2(m+4)} +-\frac{2b(ax+b)^{(m+2)/2}}{a^2(m+2)}$$ +<<*>>= +)clear all + +--S 84 of 92 +aa:=integrate(x*(a*x+b)^(m/2),x) +--R +--R +--R m log(a x + b) +--R -------------- +--R 2 2 2 2 2 +--R ((2a m + 4a )x + 2a b m x - 4b )%e +--R (1) ------------------------------------------------- +--R 2 2 2 2 +--R a m + 6a m + 8a +--R Type: Union(Expression Integer,...) +--E +@ +<<*>>= +--S 85 of 92 +bb:=(2*(a*x+b)^((m+4)/2))/(a^2*(m+4))-(2*b*(a*x+b)^((m+2)/2))/(a^2*(m+2)) +--R +--R +--R m + 4 m + 2 +--R ----- ----- +--R 2 2 +--R (2m + 4)(a x + b) + (- 2b m - 8b)(a x + b) +--R (2) ---------------------------------------------------- +--R 2 2 2 2 +--R a m + 6a m + 8a +--R Type: Expression Integer +--E +@ +<<*>>= +--S 86 of 92 +cc:=aa-bb +--R +--R +--R (3) +--R m log(a x + b) +--R -------------- +--R 2 2 2 2 2 +--R ((2a m + 4a )x + 2a b m x - 4b )%e +--R + +--R m + 4 m + 2 +--R ----- ----- +--R 2 2 +--R (- 2m - 4)(a x + b) + (2b m + 8b)(a x + b) +--R / +--R 2 2 2 2 +--R a m + 6a m + 8a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.101~~~~~$\displaystyle\int{x^2(ax+b)^{m/2}~dx}$} +$$\int{x^2(ax+b)^{m/2}~dx}= +\frac{2(ax+b)^{(m+6)/2}}{a^3(m+6)} +-\frac{4b(ax+b)^{(m+4)/2}}{a^3(m+4)} ++\frac{2b^2(ax+b)^{(m+2)/2}}{a^3(m+2)}$$ +<<*>>= +)clear all + +--S 87 of 92 +aa:=integrate(x^2*(a*x+b)^(m/2),x) +--R +--R +--R (1) +--R 3 2 3 3 3 2 2 2 2 2 3 +--R ((2a m + 12a m + 16a )x + (2a b m + 4a b m)x - 8a b m x + 16b ) +--R * +--R m log(a x + b) +--R -------------- +--R 2 +--R %e +--R / +--R 3 3 3 2 3 3 +--R a m + 12a m + 44a m + 48a +--R Type: Union(Expression Integer,...) +--E +@ +<<*>>= +--S 88 of 92 +bb:=(2*(a*x+b)^((m+6)/2))/(a^3*(m+6))-_ + (4*b*(a*x+b)^((m+4)/2))/(a^3*(m+4))+_ + (2*b^2*(a*x+b)^((m+2)/2))/(a^3*(m+2)) +--R +--R +--R (2) +--R m + 6 m + 4 +--R ----- ----- +--R 2 2 2 2 +--R (2m + 12m + 16)(a x + b) + (- 4b m - 32b m - 48b)(a x + b) +--R + +--R m + 2 +--R ----- +--R 2 2 2 2 2 +--R (2b m + 20b m + 48b )(a x + b) +--R / +--R 3 3 3 2 3 3 +--R a m + 12a m + 44a m + 48a +--R Type: Expression Integer +--E +@ +<<*>>= +--S 89 of 92 +cc:=aa-bb +--R +--R +--R (3) +--R 3 2 3 3 3 2 2 2 2 2 3 +--R ((2a m + 12a m + 16a )x + (2a b m + 4a b m)x - 8a b m x + 16b ) +--R * +--R m log(a x + b) +--R -------------- +--R 2 +--R %e +--R + +--R m + 6 m + 4 +--R ----- ----- +--R 2 2 2 2 +--R (- 2m - 12m - 16)(a x + b) + (4b m + 32b m + 48b)(a x + b) +--R + +--R m + 2 +--R ----- +--R 2 2 2 2 2 +--R (- 2b m - 20b m - 48b )(a x + b) +--R / +--R 3 3 3 2 3 3 +--R a m + 12a m + 44a m + 48a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.102~~~~~$\displaystyle\int{\frac{(ax+b)^{m/2}}{x}~dx}$} +$$\int{\frac{(ax+b)^{m/2}}{x}~dx}= +\frac{2(ax+b)^{m/2}}{m} ++b~\int{\frac{(ax+b)^{(m-2)/2}}{x}~dx}$$ +<<*>>= +)clear all + +--S 90 of 92 +aa:=integrate((a*x+b)^(m/2)/x,x) +--R +--R +--R m +--R - +--R x 2 +--I ++ (b + %L a) +--I (1) | ----------- d%L +--I ++ %L +--R Type: Union(Expression Integer,...) +--E +@ +\section{\cite{1}:14.103~~~~~$\displaystyle +\int{\frac{(ax+b)^{m/2}}{x^2}~dx}$} +$$\int{\frac{(ax+b)^{m/2}}{x^2}~dx}= +-\frac{(ax+b)^{(m+2)/2}}{bx} ++\frac{ma}{2b}~\int{\frac{(ax+b)^{m/2}}{x}~dx}$$ +<<*>>= +)clear all + +--S 91 of 92 +aa:=integrate((a*x+b)^(m/2)/x^2,x) +--R +--R +--R m +--R - +--R x 2 +--I ++ (b + %L a) +--I (1) | ----------- d%L +--R ++ 2 +--I %L +--R Type: Union(Expression Integer,...) +--E +@ +\section{\cite{1}:14.104~~~~~$\displaystyle +\int{\frac{dx}{x(ax+b)^{m/2}}}$} +$$\int{\frac{dx}{x(ax+b)^{m/2}}}= +\frac{2}{(m-2)b(ax+b)^{(m-2)/2}} ++\frac{1}{b}~\int{\frac{dx}{x(ax+b)^{(m-2)/2}}}$$ +<<*>>= +)clear all + +--S 92 of 92 +aa:=integrate(1/(x*(a*x+b)^(m/2)),x) +--R +--R +--R x +--R ++ 1 +--I (1) | -------------- d%L +--R ++ m +--R - +--R 2 +--I %L (b + %L a) +--R Type: Union(Expression Integer,...) +--E +@ + +<<*>>= +)spool +)lisp (bye) +@ + +\eject +\begin{thebibliography}{99} +\bibitem{1} Spiegel, Murray R. +{\sl Mathematical Handbook of Formulas and Tables}\\ +Schaum's Outline Series McGraw-Hill 1968 pp61-62 +\end{thebibliography} +\end{document} diff --git a/src/input/schaum3.input.pamphlet b/src/input/schaum3.input.pamphlet new file mode 100644 index 0000000..e273509 --- /dev/null +++ b/src/input/schaum3.input.pamphlet @@ -0,0 +1,409 @@ +\documentclass{article} +\usepackage{axiom} +\begin{document} +\title{\$SPAD/input schaum3.input} +\author{Timothy Daly} +\maketitle +\eject +\tableofcontents +\eject +\section{\cite{1}:14.105~~~~~$\displaystyle\int{\frac{dx}{(ax+b)(px+q)}}$} +$$\int{\frac{dx}{(ax+b)(px+q)}}= +\frac{1}{bp-aq}~\ln\left(\frac{px+q}{ax+b}\right)$$ +<<*>>= +)spool schaum3.output +)set message test on +)set message auto off +)clear all + +--S 1 of 11 +aa:=integrate(1/((a*x+b)*(p*x+q)),x) +--R +--R +--R - log(p x + q) + log(a x + b) +--R (1) ----------------------------- +--R a q - b p +--R Type: Union(Expression Integer,...) +--E +@ +<<*>>= +--S 2 of 11 +bb:=1/(b*p-a*q)*log((p*x+q)/(a*x+b)) +--R +--R +--R p x + q +--R log(-------) +--R a x + b +--R (2) - ------------ +--R a q - b p +--R Type: Expression Integer +--E +@ +<<*>>= +--S 3 of 11 +cc:=aa-bb +--R +--R +--R p x + q +--R - log(p x + q) + log(a x + b) + log(-------) +--R a x + b +--R (3) -------------------------------------------- +--R a q - b p +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.106~~~~~$\displaystyle\int{\frac{x~dx}{(ax+b)(px+q)}}$} +$$\int{\frac{x~dx}{(ax+b)(px+q)}}= +\frac{1}{bp-aq}\left\{\frac{b}{a}~\ln(ax+b)-\frac{q}{p}~\ln(px+q)\right\}$$ +<<*>>= +)clear all + +--S 4 of 11 +aa:=integrate(x/((a*x+b)*(p*x+q)),x) +--R +--R +--R a q log(p x + q) - b p log(a x + b) +--R (1) ----------------------------------- +--R 2 2 +--R a p q - a b p +--R Type: Union(Expression Integer,...) +--E +@ +<<*>>= +--S 5 of 11 +bb:=1/(b*p-a*q)*(b/a*log(a*x+b)-q/p*log(p*x+q)) +--R +--R +--R a q log(p x + q) - b p log(a x + b) +--R (2) ----------------------------------- +--R 2 2 +--R a p q - a b p +--R Type: Expression Integer +--E +@ +<<*>>= +--S 6 of 11 +cc:=aa-bb +--R +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.107~~~~~$\displaystyle\int{\frac{dx}{(ax+b)^2(px+q)}}$} +$$\int{\frac{dx}{(ax+b)^2(px+q)}}= +\frac{1}{bp-aq} +\left\{\frac{1}{ax+b}+ +\frac{p}{bp-aq}~\ln\left(\frac{px+q}{ax+b}\right)\right\}$$ +<<*>>= +)clear all + +--S 7 of 11 +aa:=integrate(1/((a*x+b)^2*(p*x+q)),x) +--R +--R +--R (a p x + b p)log(p x + q) + (- a p x - b p)log(a x + b) - a q + b p +--R (1) ------------------------------------------------------------------- +--R 3 2 2 2 2 2 2 2 3 2 +--R (a q - 2a b p q + a b p )x + a b q - 2a b p q + b p +--R Type: Union(Expression Integer,...) +--E +@ +<<*>>= +--S 8 of 11 +bb:=1/(b*p-a*q)*(1/(a*x+b)+p/(b*p-a*q)*log((p*x+q)/(a*x+b))) +--R +--R +--R p x + q +--R (a p x + b p)log(-------) - a q + b p +--R a x + b +--R (2) ------------------------------------------------------ +--R 3 2 2 2 2 2 2 2 3 2 +--R (a q - 2a b p q + a b p )x + a b q - 2a b p q + b p +--R Type: Expression Integer +--E +@ +<<*>>= +--S 9 of 11 +cc:=aa-bb +--R +--R +--R p x + q +--R p log(p x + q) - p log(a x + b) - p log(-------) +--R a x + b +--R (3) ------------------------------------------------ +--R 2 2 2 2 +--R a q - 2a b p q + b p +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.108~~~~~$\displaystyle\int{\frac{x~dx}{(ax+b)^2(px+q)}}$} +$$\int{\frac{x~dx}{(ax+b)^2(px+q)}}= +\frac{1}{bp-aq} +\left\{\frac{q}{bp-aq} +~\ln\left(\frac{ax+b}{px+q}\right)-\frac{b}{a(ax+b)}\right\}$$ + +<<*>>= +)clear all + +--S 10 of 11 +aa:=integrate(x/((a*x+b)^2*(p*x+q)),x) +--R +--R +--R (1) +--R 2 2 2 +--R (- a q x - a b q)log(p x + q) + (a q x + a b q)log(a x + b) + a b q - b p +--R ------------------------------------------------------------------------- +--R 4 2 3 2 2 2 3 2 2 2 3 2 +--R (a q - 2a b p q + a b p )x + a b q - 2a b p q + a b p +--R Type: Union(Expression Integer,...) +--E +@ +<<*>>= +--S 11 of 11 +bb:=1/(b*p-a*q)*(q/(b*p-a*q)*log((a*x+b)/(p*x+q))-b/(a*(a*x+b))) +--R +--R +--R 2 a x + b 2 +--R (a q x + a b q)log(-------) + a b q - b p +--R p x + q +--R (2) -------------------------------------------------------- +--R 4 2 3 2 2 2 3 2 2 2 3 2 +--R (a q - 2a b p q + a b p )x + a b q - 2a b p q + a b p +--R Type: Expression Integer +--E +@ +<<*>>= +cc:=aa-bb +--R +--R +--R a x + b +--R - q log(p x + q) + q log(a x + b) - q log(-------) +--R p x + q +--R (3) -------------------------------------------------- +--R 2 2 2 2 +--R a q - 2a b p q + b p +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.109~~~~~$\displaystyle +\int{\frac{x^2~dx}{(ax+b)^2(px+q)}}$} +$$\int{\frac{x^2~dx}{(ax+b)^2(px+q)}}=$$ +$$\frac{b^2}{(bp-aq)a^2(ax+b)}+\frac{1}{(bp-aq)^2} +\left\{\frac{q^2}{p}~\ln(px+q)+\frac{b(bp-2aq)}{a^2}~\ln(ax+b)\right\}$$ +<<*>>= +)clear all + +--S +aa:=integrate(x^2/((a*x+b)^2*(p*x+q)),x) +--R +--R +--R (1) +--R 3 2 2 2 +--R (a q x + a b q )log(p x + q) +--R + +--R 2 2 2 2 3 2 2 3 2 +--R ((- 2a b p q + a b p )x - 2a b p q + b p )log(a x + b) - a b p q + b p +--R / +--R 5 2 4 2 3 2 3 4 2 3 2 2 2 3 3 +--R (a p q - 2a b p q + a b p )x + a b p q - 2a b p q + a b p +--R Type: Union(Expression Integer,...) +--E +@ +<<*>>= +--S +bb:=b^2/((b*p-a*q)*a^2*(a*x+b))+_ + 1/(b*p-a*q)^2*(q^2/p*log(p*x+q)+((b*(b*p-2*a*q))/a^2)*log(a*x+b)) +--R +--R +--R (2) +--R 3 2 2 2 +--R (a q x + a b q )log(p x + q) +--R + +--R 2 2 2 2 3 2 2 3 2 +--R ((- 2a b p q + a b p )x - 2a b p q + b p )log(a x + b) - a b p q + b p +--R / +--R 5 2 4 2 3 2 3 4 2 3 2 2 2 3 3 +--R (a p q - 2a b p q + a b p )x + a b p q - 2a b p q + a b p +--R Type: Expression Integer +--E +@ +<<*>>= +--S +cc:=aa-bb +--R +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.110~~~~~$\displaystyle\int{\frac{dx}{(ax+b)^m(px+q)^n}}$} +$$\int{\frac{dx}{(ax+b)^m(px+q)^n}}=$$ +$$\frac{-1}{(n-1)(bp-aq)} +\left\{\frac{1}{(ax+b)^{m-1}(px+q)^{n-1}}+ +a(m+n-2)~\int{\frac{dx}{(ax+b)^m(px+q)^{n-1}}}\right\}$$ +<<*>>= +)clear all + +--S +aa:=integrate(1/((a*x+b)^m*(p*x+q)^n),x) +--R +--R +--R x +--R ++ 1 +--I (1) | ---------------------- d%L +--R ++ m n +--I (b + %L a) (q + %L p) +--R Type: Union(Expression Integer,...) +--E +@ +<<*>>= +--S +dd:=integrate(1/((a*x+b)^m*(p*x+q)^(n-1)),x) +--R +--R +--R x +--R ++ 1 +--I (2) | -------------------------- d%L +--R ++ m n - 1 +--I (b + %L a) (q + %L p) +--R Type: Union(Expression Integer,...) +--E +@ + +<<*>>= +--S +bb:=-1/((n-1)*(b*p-a*q))*(1/((a*x+b)^(m-1)*(p*x+q)^(n-1))+a*(m+n-2)*dd) +--R +--R +--R (3) +--R m - 1 n - 1 +--R (a n + a m - 2a)(a x + b) (p x + q) +--R * +--R x +--R ++ 1 +--I | -------------------------- d%L +--R ++ m n - 1 +--I (b + %L a) (q + %L p) +--R + +--R 1 +--R / +--R m - 1 n - 1 +--R ((a n - a)q + (- b n + b)p)(a x + b) (p x + q) +--R Type: Expression Integer +--E +@ +<<*>>= +--S +cc:=aa-bb +--R +--R +--R (4) +--R m - 1 n - 1 +--R (- a n - a m + 2a)(a x + b) (p x + q) +--R * +--R x +--R ++ 1 +--I | -------------------------- d%L +--R ++ m n - 1 +--I (b + %L a) (q + %L p) +--R + +--R m - 1 n - 1 +--R ((a n - a)q + (- b n + b)p)(a x + b) (p x + q) +--R * +--R x +--R ++ 1 +--I | ---------------------- d%L +--R ++ m n +--I (b + %L a) (q + %L p) +--R + +--R - 1 +--R / +--R m - 1 n - 1 +--R ((a n - a)q + (- b n + b)p)(a x + b) (p x + q) +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.111~~~~~$\displaystyle\int{\frac{ax+b}{px+q}~dx}$} +$$\int{\frac{ax+b}{px+q}~dx}=\frac{ax}{p}+\frac{bp-aq}{p^2}~\ln(px+q)$$ +<<*>>= +)clear all + +--S +aa:=integrate((a*x+b)/(p*x+q),x) +--R +--R +--R (- a q + b p)log(p x + q) + a p x +--R (1) --------------------------------- +--R 2 +--R p +--R Type: Union(Expression Integer,...) +--E +@ +<<*>>= +--S +bb:=(a*x)/p+(b*p-a*q)/p^2*log(p*x+q) +--R +--R +--R (- a q + b p)log(p x + q) + a p x +--R (2) --------------------------------- +--R 2 +--R p +--R Type: Expression Integer +--E +@ +<<*>>= +--S +cc:=aa-bb +--R +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.112~~~~~$\displaystyle\int{\frac{(ax+b)^m}{(px+q)^n}~dx}$} +$$\int{\frac{(ax+b)^m}{(px+q)^n}~dx}=\left\{ +\begin{array}{c} +\frac{-1}{(n-1)(bp-aq)} +\left\{\frac{(ax+b)^{m+1}}{(px+q)^{n-1}}+(n-m-2)a +\int{\frac{(ax+b)^m}{(px+q)^{n-1}}}~dx\right\}\\ +\frac{-1}{(n-m-1)p}+\left\{\frac{(ax+b)^m}{(px+q)^{n-1}}+m(bp-aq) +\int{\frac{(ax+b)^{m-1}}{(px+q)^n}}~dx\right\}\\ +\frac{-1}{(n-1)p}\left\{\frac{(ax+b)^m}{(px+q)^{n-1}}-ma +\int{\frac{(ax+b)^{m-1}}{(px+q)^{n-1}}}~dx\right\} +\end{array} +\right.$$ +<<*>>= +)clear all + +--S +aa:=integrate((a*x+b)^m/(p*x+q)^n,x) +--R +--R +--R x m +--I ++ (b + %L a) +--I (1) | ----------- d%L +--R ++ n +--I (q + %L p) +--R Type: Union(Expression Integer,...) +--R +--E +<<*>>= +)spool +)lisp (bye) +@ + +\eject +\begin{thebibliography}{99} +\bibitem{1} Spiegel, Murray R. +{\sl Mathematical Handbook of Formulas and Tables}\\ +Schaum's Outline Series McGraw-Hill 1968 pp62-63 +\end{thebibliography} +\end{document} diff --git a/src/input/schaum4.input.pamphlet b/src/input/schaum4.input.pamphlet new file mode 100644 index 0000000..b57e857 --- /dev/null +++ b/src/input/schaum4.input.pamphlet @@ -0,0 +1,212 @@ +\documentclass{article} +\usepackage{axiom} +\begin{document} +\title{\$SPAD/input schaum4.input} +\author{Timothy Daly} +\maketitle +\eject +\tableofcontents +\eject +\section{\cite{1}:14.113~~~~~$\displaystyle\int{\frac{px+q}{\sqrt{ax+b}}}~dx$} +$$\int{\frac{px+q}{\sqrt{ax+b}}}= +\frac{2(apx+3aq-2bp)}{3a^2}\sqrt{ax+b}$$ +<<*>>= +)spool schaum4.output +)set message test on +)set message auto off +)clear all + +--S 1 of 7 +aa:=integrate((p*x+q)/sqrt(a*x+b),x) +--R +--R +--R +-------+ +--R (2a p x + 6a q - 4b p)\|a x + b +--R (1) -------------------------------- +--R 2 +--R 3a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.114~~~~~$\displaystyle +\int{\frac{dx}{(px+q)\sqrt{ax+b}}}~dx$} +$$\int{\frac{dx}{(px+q)\sqrt{ax+b}}}= +\left\{ +\begin{array}{l} +\frac{1}{\sqrt{bp-aq}\sqrt{p}}\ln\left( +\frac{\sqrt{p(ax+b)}-\sqrt{bp-aq}}{\sqrt{p(ax+b)}+\sqrt{bp-aq}}\right)\\ +\frac{2}{\sqrt{aq-bp}\sqrt{p}}\tan^{-1}\sqrt{\frac{p(ax+b)}{aq-bp}} +\end{array} +\right. +$$ +<<*>>= +)clear all + +--S 2 of 7 +aa:=integrate(1/((p*x+q)*sqrt(a*x+b)),x) +--R +--R +--R (1) +--R +--------------+ +--R 2 +-------+ | 2 +--R (2a p q - 2b p )\|a x + b + (a p x - a q + 2b p)\|- a p q + b p +--R log(------------------------------------------------------------------) +--R p x + q +--R [-----------------------------------------------------------------------, +--R +--------------+ +--R | 2 +--R \|- a p q + b p +--R +------------+ +--R | 2 +-------+ +--R \|a p q - b p \|a x + b +--R 2atan(-------------------------) +--R a q - b p +--R --------------------------------] +--R +------------+ +--R | 2 +--R \|a p q - b p +--R Type: Union(List Expression Integer,...) +--E +@ + +\section{\cite{1}:14.115~~~~~$\displaystyle\int{\frac{\sqrt{ax+b}}{px+q}}~dx$} +$$\int{\frac{\sqrt{ax+b}}{px+q}}= +\left\{ +\begin{array}{l} +\frac{2\sqrt{ax+b}}{p}+\frac{\sqrt{bp-aq}}{p\sqrt{p}}\ln\left( +\frac{\sqrt{p(ax+b)}-\sqrt{bp-aq}}{\sqrt{p(ax+b)}+\sqrt{bp-aq}}\right)\\ +\frac{2\sqrt{ax+b}}{p}-\frac{2\sqrt{aq-bp}}{p\sqrt{p}} +\tan^{-1}\sqrt{\frac{p(ax+b)}{aq-bp}} +\end{array} +\right.$$ +<<*>>= +)clear all + +--S 3 of 7 +aa:=integrate(sqrt(a*x+b)/(p*x+q),x) +--R +--R +--R (1) +--R [ +--R +-----------+ +--R |- a q + b p +-------+ +--R +-----------+ - 2p |----------- \|a x + b + a p x - a q + 2b p +--R |- a q + b p \| p +--R |----------- log(-------------------------------------------------) +--R \| p p x + q +--R + +--R +-------+ +--R 2\|a x + b +--R / +--R p +--R , +--R +---------+ +-------+ +--R |a q - b p \|a x + b +-------+ +--R - 2 |--------- atan(------------ + 2\|a x + b +--R \| p +---------+ +--R |a q - b p +--R |--------- +--R \| p +--R -----------------------------------------------] +--R p +--R Type: Union(List Expression Integer,...) +--E +@ + +\section{\cite{1}:14.116~~~~~$\displaystyle\int{(px+b)^n\sqrt{ax+b}}~dx$} +$$\int{(px+b)^n\sqrt{ax+b}}= +\frac{2(px+q)^{n+1}\sqrt{ax+b}}{(2n+3)p}+\frac{bp-aq}{(2n+3)p} +\int{\frac{(px+q)^n}{\sqrt{ax+b}}}~dx$$ + +<<*>>= +)clear all + +--S 4 of 7 +aa:=integrate((p*x+q)^n*sqrt(a*x+b),x) +--R +--R +--R x +--R ++ n +--------+ +--I (1) | (q + %L p) \|b + %L a d%L +--R ++ +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.117~~~~~$\displaystyle +\int{\frac{dx}{(px+b)^n\sqrt{ax+b}}}$} +$$\int{\frac{dx}{(px+b)^n\sqrt{ax+b}}}= +\frac{\sqrt{ax+b}}{(n-1)(aq-bp)(px+q)^{n-1}}+ +\frac{(2n-3)a}{2(n-1)(aq-bp)} +\int{\frac{dx}{(px+q)^{n-1}\sqrt{ax+b}}}$$ + +<<*>>= +)clear all + +--S 5 of 7 +aa:=integrate(1/((p*x+q)^n*sqrt(a*x+b)),x) +--R +--R +--R x +--R ++ 1 +--I (1) | ---------------------- d%L +--R ++ n +--------+ +--I (q + %L p) \|b + %L a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.118~~~~~$\displaystyle +\int{\frac{(px+q)^n}{\sqrt{ax+b}}}~dx$} +$$\int{\frac{(px+q)^n}{\sqrt{ax+b}}}= +\frac{2(px+q)^n\sqrt{ax+b}}{(2n+1)a}+ +\frac{2n(aq-bp)}{(2n+1)a} +\int{\frac{(px+q)^{n-1}}{\sqrt{ax+b}}}$$ +<<*>>= +)clear all + +--S 6 of 7 +aa:=integrate((p*x+q)^n/sqrt(a*x+b),x) +--R +--R +--R x n +--I ++ (q + %L p) +--I (1) | ----------- d%L +--R ++ +--------+ +--I \|b + %L a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.119~~~~~$\displaystyle +\int{\frac{\sqrt{ax+b}}{(px+q)^n}}~dx$} +$$\int{\frac{\sqrt{ax+b}}{(px+q)^n}}= +\frac{-\sqrt{ax+b}}{(n-1)p(px+q)^{n-1}}+ +\frac{a}{2(n-1)p}\int{\frac{dx}{(px+q)^{n-1}\sqrt{ax+b}}}$$ +<<*>>= +)clear all + +--S 7 of 7 +aa:=integrate(sqrt(a*x+b)/(p*x+q)^n,x) +--R +--R +--R x +--------+ +--I ++ \|b + %L a +--I (1) | ----------- d%L +--R ++ n +--I (q + %L p) +--R Type: Union(Expression Integer,...) +--E + +)spool +)lisp (bye) +@ + +\eject +\begin{thebibliography}{99} +\bibitem{1} Spiegel, Murray R. +{\sl Mathematical Handbook of Formulas and Tables}\\ +Schaum's Outline Series McGraw-Hill 1968 p63 +\end{thebibliography} +\end{document} diff --git a/src/input/schaum5.input.pamphlet b/src/input/schaum5.input.pamphlet new file mode 100644 index 0000000..a784b92 --- /dev/null +++ b/src/input/schaum5.input.pamphlet @@ -0,0 +1,367 @@ +\documentclass{article} +\usepackage{axiom} +\begin{document} +\title{\$SPAD/input schaum5.input} +\author{Timothy Daly} +\maketitle +\eject +\tableofcontents +\eject +\section{\cite{1}:14.120~~~~~$\displaystyle +\int{\frac{dx}{\sqrt{(ax+b)(px+q)}}}$} +$$\int{\frac{dx}{\sqrt{(ax+b)(px+q)}}}= +\left\{ +\begin{array}{l} +\frac{2}{\sqrt{ap}}\ln\left(\sqrt{a(px+q)}+\sqrt{p(ax+b)}\right)\\ +\frac{2}{\sqrt{-ap}}\tan^{-1}\sqrt{\frac{-p(ax+b)}{a(px+b)}} +\end{array} +\right.$$ +<<*>>= +)spool schaum5.output +)set message test on +)set message auto off +)clear all + +--S 1 of 5 +aa:=integrate(1/sqrt((a*x+b)*(p*x+q)),x) +--R +--R +--R (1) +--R [ +--R log +--R +---------------------------+ +--R +---+ +---+ | 2 +--R (2\|a p \|b q - 2a p x)\|a p x + (a q + b p)x + b q +--R + +--R +---+ 2 +---+ +--R 2a p x\|b q + (- 2a p x + (- a q - b p)x - 2b q)\|a p +--R / +--R +---------------------------+ +--R +---+ | 2 +--R 2\|b q \|a p x + (a q + b p)x + b q + (- a q - b p)x - 2b q +--R / +--R +---+ +--R \|a p +--R , +--R +---------------------------+ +--R +-----+ | 2 +-----+ +---+ +--R \|- a p \|a p x + (a q + b p)x + b q - \|- a p \|b q +--R 2atan(-------------------------------------------------------) +--R a p x +--R --------------------------------------------------------------] +--R +-----+ +--R \|- a p +--R Type: Union(List Expression Integer,...) +--E +@ + +\section{\cite{1}:14.121~~~~~$\displaystyle +\int{\frac{x~dx}{\sqrt{(ax+b)(px+q)}}}$} +$$\int{\frac{x~dx}{\sqrt{(ax+b)(px+q)}}}= +\frac{\sqrt{(ax+b)(px+q)}}{ap}-\frac{bp+aq}{2ap} +\int{\frac{dx}{\sqrt{(ax+b)(px+q)}}} +$$ +<<*>>= +)clear all + +--S 2 of 5 +aa:=integrate(x/sqrt((a*x+b)*(p*x+q)),x) +--R +--R +--R (1) +--R [ +--R +---------------------------+ +--R +---+ | 2 +--R (2a q + 2b p)\|b q \|a p x + (a q + b p)x + b q +--R + +--R 2 2 2 2 2 2 +--R (- a q - 2a b p q - b p )x - 2a b q - 2b p q +--R * +--R log +--R +---------------------------+ +--R +---+ +---+ | 2 +--R (2\|a p \|b q + 2a p x)\|a p x + (a q + b p)x + b q +--R + +--R +---+ 2 +---+ +--R - 2a p x\|b q + (- 2a p x + (- a q - b p)x - 2b q)\|a p +--R / +--R +---------------------------+ +--R +---+ | 2 +--R 2\|b q \|a p x + (a q + b p)x + b q + (- a q - b p)x - 2b q +--R + +--R +---------------------------+ +--R +---+ | 2 +--R (- 2a q - 2b p)x\|a p \|a p x + (a q + b p)x + b q +--R + +--R 2 +---+ +---+ +--R (4a p x + (2a q + 2b p)x)\|a p \|b q +--R / +--R +---------------------------+ +--R +---+ +---+ | 2 +--R 4a p\|a p \|b q \|a p x + (a q + b p)x + b q +--R + +--R 2 2 +---+ +--R ((- 2a p q - 2a b p )x - 4a b p q)\|a p +--R , +--R +--R +---------------------------+ +--R +---+ | 2 +--R (- 2a q - 2b p)\|b q \|a p x + (a q + b p)x + b q +--R + +--R 2 2 2 2 2 2 +--R (a q + 2a b p q + b p )x + 2a b q + 2b p q +--R * +--R +---------------------------+ +--R +-----+ | 2 +-----+ +---+ +--R \|- a p \|a p x + (a q + b p)x + b q - \|- a p \|b q +--R atan(-------------------------------------------------------) +--R a p x +--R + +--R +---------------------------+ +--R +-----+ | 2 +--R (- a q - b p)x\|- a p \|a p x + (a q + b p)x + b q +--R + +--R 2 +-----+ +---+ +--R (2a p x + (a q + b p)x)\|- a p \|b q +--R / +--R +---------------------------+ +--R +-----+ +---+ | 2 +--R 2a p\|- a p \|b q \|a p x + (a q + b p)x + b q +--R + +--R 2 2 +-----+ +--R ((- a p q - a b p )x - 2a b p q)\|- a p +--R ] +--R Type: Union(List Expression Integer,...) +--E +@ + +\section{\cite{1}:14.122~~~~~$\displaystyle\int{\sqrt{(ax+b)(px+q)}}~dx$} +$$\int{\sqrt{(ax+b)(px+q)}}= +\frac{2apx+bp+aq}{4ap}\sqrt{(ax+b)(px+q)}- +\frac{(bp-aq)^2}{8ap}\int{\frac{dx}{\sqrt{(ax+b)(px+q)}}} +$$ +<<*>>= +)clear all + +--S 3 of 5 +aa:=integrate(sqrt((a*x+b)*(p*x+q)),x) +--R +--R +--R (1) +--R [ +--R 3 3 2 2 2 2 3 3 2 3 2 2 +--R (4a q - 4a b p q - 4a b p q + 4b p )x + 8a b q - 16a b p q +--R + +--R 3 2 +--R 8b p q +--R * +--R +---------------------------+ +--R +---+ | 2 +--R \|b q \|a p x + (a q + b p)x + b q +--R + +--R 4 4 3 3 2 2 2 2 3 3 4 4 2 +--R (- a q - 4a b p q + 10a b p q - 4a b p q - b p )x +--R + +--R 3 4 2 2 3 3 2 2 4 3 2 2 4 +--R (- 8a b q + 8a b p q + 8a b p q - 8b p q)x - 8a b q +--R + +--R 3 3 4 2 2 +--R 16a b p q - 8b p q +--R * +--R log +--R +---------------------------+ +--R +---+ +---+ | 2 +--R (2\|a p \|b q + 2a p x)\|a p x + (a q + b p)x + b q +--R + +--R +---+ 2 +---+ +--R - 2a p x\|b q + (- 2a p x + (- a q - b p)x - 2b q)\|a p +--R / +--R +---------------------------+ +--R +---+ | 2 +--R 2\|b q \|a p x + (a q + b p)x + b q + (- a q - b p)x - 2b q +--R + +--R 3 2 2 2 2 3 3 +--R (- 4a p q - 24a b p q - 4a b p )x +--R + +--R 3 3 2 2 2 2 3 3 2 +--R (- 2a q - 46a b p q - 46a b p q - 2b p )x +--R + +--R 2 3 2 2 3 2 +--R (- 8a b q - 48a b p q - 8b p q)x +--R * +--R +---------------------------+ +--R +---+ | 2 +--R \|a p \|a p x + (a q + b p)x + b q +--R + +--R 3 2 2 3 4 3 2 2 2 2 3 3 +--R (16a p q + 16a b p )x + (24a p q + 80a b p q + 24a b p )x +--R + +--R 3 3 2 2 2 2 3 3 2 +--R (6a q + 74a b p q + 74a b p q + 6b p )x +--R + +--R 2 3 2 2 3 2 +--R (8a b q + 48a b p q + 8b p q)x +--R * +--R +---+ +---+ +--R \|a p \|b q +--R / +--R 2 2 +---+ +---+ +--R ((32a p q + 32a b p )x + 64a b p q)\|a p \|b q +--R * +--R +---------------------------+ +--R | 2 +--R \|a p x + (a q + b p)x + b q +--R + +--R 3 2 2 2 2 3 2 2 2 2 2 +--R (- 8a p q - 48a b p q - 8a b p )x + (- 64a b p q - 64a b p q)x +--R + +--R 2 2 +--R - 64a b p q +--R * +--R +---+ +--R \|a p +--R , +--R +--R 3 3 2 2 2 2 3 3 2 3 +--R (- 4a q + 4a b p q + 4a b p q - 4b p )x - 8a b q +--R + +--R 2 2 3 2 +--R 16a b p q - 8b p q +--R * +--R +---------------------------+ +--R +---+ | 2 +--R \|b q \|a p x + (a q + b p)x + b q +--R + +--R 4 4 3 3 2 2 2 2 3 3 4 4 2 +--R (a q + 4a b p q - 10a b p q + 4a b p q + b p )x +--R + +--R 3 4 2 2 3 3 2 2 4 3 2 2 4 3 3 +--R (8a b q - 8a b p q - 8a b p q + 8b p q)x + 8a b q - 16a b p q +--R + +--R 4 2 2 +--R 8b p q +--R * +--R +---------------------------+ +--R +-----+ | 2 +-----+ +---+ +--R \|- a p \|a p x + (a q + b p)x + b q - \|- a p \|b q +--R atan(-------------------------------------------------------) +--R a p x +--R + +--R 3 2 2 2 2 3 3 +--R (- 2a p q - 12a b p q - 2a b p )x +--R + +--R 3 3 2 2 2 2 3 3 2 +--R (- a q - 23a b p q - 23a b p q - b p )x +--R + +--R 2 3 2 2 3 2 +--R (- 4a b q - 24a b p q - 4b p q)x +--R * +--R +---------------------------+ +--R +-----+ | 2 +--R \|- a p \|a p x + (a q + b p)x + b q +--R + +--R 3 2 2 3 4 3 2 2 2 2 3 3 +--R (8a p q + 8a b p )x + (12a p q + 40a b p q + 12a b p )x +--R + +--R 3 3 2 2 2 2 3 3 2 +--R (3a q + 37a b p q + 37a b p q + 3b p )x +--R + +--R 2 3 2 2 3 2 +--R (4a b q + 24a b p q + 4b p q)x +--R * +--R +-----+ +---+ +--R \|- a p \|b q +--R / +--R 2 2 +-----+ +---+ +--R ((16a p q + 16a b p )x + 32a b p q)\|- a p \|b q +--R * +--R +---------------------------+ +--R | 2 +--R \|a p x + (a q + b p)x + b q +--R + +--R 3 2 2 2 2 3 2 2 2 2 2 +--R (- 4a p q - 24a b p q - 4a b p )x + (- 32a b p q - 32a b p q)x +--R + +--R 2 2 +--R - 32a b p q +--R * +--R +-----+ +--R \|- a p +--R ] +--R Type: Union(List Expression Integer,...) +--E +@ + +\section{\cite{1}:14.123~~~~~$\displaystyle\int{\sqrt{\frac{px+q}{ax+b}}}~dx$} +$$\int{\sqrt{\frac{px+q}{ax+b}}}= +\frac{\sqrt{(ax+b)(px+q)}}{a}+\frac{aq-bp}{2a} +\int{\frac{dx}{\sqrt{(ax+b)(px+q)}}} +$$ +<<*>>= +)clear all + +--S 4 of 5 +aa:=integrate(sqrt((p*x+q)/(a*x+b)),x) +--R +--R +--R (1) +--R [ +--R (a q - b p) +--R * +--R +-------+ +--R +---+ 2 |p x + q +--R log((2a p x + a q + b p)\|a p + (2a p x + 2a b p) |------- ) +--R \|a x + b +--R + +--R +-------+ +--R |p x + q +---+ +--R (2a x + 2b) |------- \|a p +--R \|a x + b +--R / +--R +---+ +--R 2a\|a p +--R , +--R +-------+ +--R +-----+ |p x + q +--R \|- a p |------- +-------+ +--R \|a x + b +-----+ |p x + q +--R (a q - b p)atan(------------------) + (a x + b)\|- a p |------- +--R p \|a x + b +--R -----------------------------------------------------------------] +--R +-----+ +--R a\|- a p +--R Type: Union(List Expression Integer,...) +--E +@ + +\section{\cite{1}:14.124~~~~~$\displaystyle +\int{\frac{dx}{(px+q)\sqrt{(ax+b)(px+q)}}}~dx$} +$$\int{\frac{dx}{(px+q)\sqrt{(ax+b)(px+q)}}}= +\frac{2\sqrt{ax+b}}{(aq-bp)\sqrt{px+q}} +$$ +<<*>>= +)clear all + +--S 5 of 5 +aa:=integrate(1/((p*x+q)*sqrt((a*x+b)*(p*x+q))),x) +--R +--R +--R 2x +--R (1) --------------------------------------------------- +--R +---------------------------+ +--R | 2 +---+ +--R q\|a p x + (a q + b p)x + b q + (- p x - q)\|b q +--R Type: Union(Expression Integer,...) +--E + +)spool +)lisp (bye) +@ + +\eject +\begin{thebibliography}{99} +\bibitem{1} Spiegel, Murray R. +{\sl Mathematical Handbook of Formulas and Tables}\\ +Schaum's Outline Series McGraw-Hill 1968 pp63-64 +\end{thebibliography} +\end{document} diff --git a/src/input/schaum6.input.pamphlet b/src/input/schaum6.input.pamphlet new file mode 100644 index 0000000..1a4b430 --- /dev/null +++ b/src/input/schaum6.input.pamphlet @@ -0,0 +1,400 @@ +\documentclass{article} +\usepackage{axiom} +\begin{document} +\title{\$SPAD/input schaum6.input} +\author{Timothy Daly} +\maketitle +\eject +\tableofcontents +\eject +\section{\cite{1}:14.125~~~~~$\displaystyle\int{\frac{dx}{x^2+a^2}}$} +$$\int{\frac{dx}{x^2+a^2}}=\frac{1}{a}\tan^{-1}\frac{x}{a}$$ +<<*>>= +)spool schaum6.output +)set message test on +)set message auto off +)clear all + +--S 1 of 19 +aa:=integrate(1/(x^2+a^2),x) +--R +--R +--R x +--R atan(-) +--R a +--R (1) ------- +--R a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.126~~~~~$\displaystyle\int{\frac{x~dx}{x^2+a^2}}$} +$$\int{\frac{x~dx}{x^2+a^2}}=\frac{1}{2}\ln(x^2+a^2)$$ +<<*>>= +)clear all + +--S 2 of 19 +aa:=integrate(x/(x^2+a^2),x) +--R +--R +--R 2 2 +--R log(x + a ) +--R (1) ------------ +--R 2 +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.127~~~~~$\displaystyle\int{\frac{x^2~dx}{x^2+a^2}}$} +$$\int{\frac{x^2~dx}{x^2+a^2}}=x-a\tan^{-1}\frac{x}{a}$$ +<<*>>= +)clear all + +--S 3 of 19 +aa:=integrate(x^2/(x^2+a^2),x) +--R +--R +--R x +--R (1) - a atan(-) + x +--R a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.128~~~~~$\displaystyle\int{\frac{x^3~dx}{x^2+a^2}}$} +$$\int{\frac{x^3~dx}{x^2+a^2}}=\frac{x^2}{2}-\frac{a^2}{2}\ln(x^2+a^2)$$ + +<<*>>= +)clear all + +--S 4 of 19 +aa:=integrate(x^3/(x^2+a^2),x) +--R +--R +--R 2 2 2 2 +--R - a log(x + a ) + x +--R (1) --------------------- +--R 2 +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.129~~~~~$\displaystyle\int{\frac{dx}{x(x^2+a^2)}}~dx$} +$$\int{\frac{dx}{x(x^2+a^2)}}= +\frac{1}{2a^2}\ln\left(\frac{x^2}{x^2+a^2}\right) +$$ +<<*>>= +)clear all + +--S 5 of 19 +aa:=integrate(1/(x*(x^2+a^2)),x) +--R +--R +--R 2 2 +--R - log(x + a ) + 2log(x) +--R (1) ------------------------ +--R 2 +--R 2a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.130~~~~~$\displaystyle\int{\frac{dx}{x^2(x^2+a^2)}}~dx$} +$$\int{\frac{dx}{x^2(x^2+a^2)}}= +-\frac{1}{a^2x}-\frac{1}{a^3}\tan^{-1}\frac{x}{a} +$$ +<<*>>= +)clear all + +--S 6 of 19 +aa:=integrate(1/(x^2*(x^2+a^2)),x) +--R +--R +--R x +--R - x atan(-) - a +--R a +--R (1) --------------- +--R 3 +--R a x +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.131~~~~~$\displaystyle\int{\frac{dx}{x^3(x^2+a^2)}}~dx$} +$$\int{\frac{dx}{x^3(x^2+a^2)}}= +-\frac{1}{2a^2x^2}-\frac{1}{2a^4}\ln\left(\frac{x^2}{x^2+a^2}\right) +$$ +<<*>>= +)clear all + +--S 7 of 19 +aa:=integrate(1/(x^3*(x^2+a^2)),x) +--R +--R +--R 2 2 2 2 2 +--R x log(x + a ) - 2x log(x) - a +--R (1) ------------------------------- +--R 4 2 +--R 2a x +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.132~~~~~$\displaystyle\int{\frac{dx}{(x^2+a^2)^2}}~dx$} +$$\int{\frac{dx}{(x^2+a^2)^2}}= +\frac{x}{2a^2(x^2+a^2)}+\frac{1}{2a^3}\tan^{-1}\frac{x}{a} +$$ +<<*>>= +)clear all + +--S 8 of 19 +aa:=integrate(1/((x^2+a^2)^2),x) +--R +--R +--R 2 2 x +--R (x + a )atan(-) + a x +--R a +--R (1) ---------------------- +--R 3 2 5 +--R 2a x + 2a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.133~~~~~$\displaystyle\int{\frac{x~dx}{(x^2+a^2)^2}}~dx$} +$$\int{\frac{x~dx}{(x^2+a^2)^2}}= +\frac{-1}{2(x^2+a^2)} +$$ +<<*>>= +)clear all + +--S 9 of 19 +aa:=integrate(x/((x^2+a^2)^2),x) +--R +--R +--R 1 +--R (1) - --------- +--R 2 2 +--R 2x + 2a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.134~~~~~$\displaystyle\int{\frac{x^2dx}{(x^2+a^2)^2}}~dx$} +$$\int{\frac{x^2dx}{(x^2+a^2)^2}}= +\frac{-x}{2(x^2+a^2)}+\frac{1}{2a}\tan^{-1}\frac{x}{a} +$$ +<<*>>= +)clear all + +--S 10 of 19 +aa:=integrate(x^2/((x^2+a^2)^2),x) +--R +--R +--R 2 2 x +--R (x + a )atan(-) - a x +--R a +--R (1) ---------------------- +--R 2 3 +--R 2a x + 2a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.135~~~~~$\displaystyle\int{\frac{x^3dx}{(x^2+a^2)^2}}~dx$} +$$\int{\frac{x^3dx}{(x^2+a^2)^2}}= +\frac{a^2}{2(x^2+a^2)}+\frac{1}{2}\ln(x^2+a^2) +$$ +<<*>>= +)clear all + +--S 11 of 19 +aa:=integrate(x^3/((x^2+a^2)^2),x) +--R +--R +--R 2 2 2 2 2 +--R (x + a )log(x + a ) + a +--R (1) -------------------------- +--R 2 2 +--R 2x + 2a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.136~~~~~$\displaystyle\int{\frac{dx}{x(x^2+a^2)^2}}~dx$} +$$\int{\frac{dx}{x(x^2+a^2)^2}}= +\frac{1}{2a^2(x^2+a^2)}+\frac{1}{2a^4}\ln\left(\frac{x^2}{x^2+a^2}\right) +$$ +<<*>>= +)clear all + +--S 12 of 19 +aa:=integrate(1/(x*(x^2+a^2)^2),x) +--R +--R +--R 2 2 2 2 2 2 2 +--R (- x - a )log(x + a ) + (2x + 2a )log(x) + a +--R (1) ------------------------------------------------ +--R 4 2 6 +--R 2a x + 2a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.137~~~~~$\displaystyle\int{\frac{dx}{x^2(x^2+a^2)^2}}~dx$} +$$\int{\frac{dx}{x^2(x^2+a^2)^2}}= +-\frac{1}{a^4x}-\frac{x}{2a^4(x^2+a^2)}-\frac{3}{2a^5}\tan^{-1}\frac{x}{a} +$$ +<<*>>= +)clear all + +--S 13 of 19 +aa:=integrate(1/((x^2+a^2)^2),x) +--R +--R +--R 2 2 x +--R (x + a )atan(-) + a x +--R a +--R (1) ---------------------- +--R 3 2 5 +--R 2a x + 2a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.138~~~~~$\displaystyle\int{\frac{dx}{x^3(x^2+a^2)^2}}~dx$} +$$\int{\frac{dx}{x^3(x^2+a^2)^2}}= +-\frac{1}{2a^4x^2}-\frac{1}{2a^4(x^2+a^2)}- +\frac{1}{a^6}\ln\left(\frac{x^2}{x^2+a^2}\right) +$$ +<<*>>= +)clear all + +--S 14 of 19 +aa:=integrate(1/(x^3*(x^2+a^2)^2),x) +--R +--R +--R 4 2 2 2 2 4 2 2 2 2 4 +--R (2x + 2a x )log(x + a ) + (- 4x - 4a x )log(x) - 2a x - a +--R (1) -------------------------------------------------------------- +--R 6 4 8 2 +--R 2a x + 2a x +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.139~~~~~$\displaystyle\int{\frac{dx}{(x^2+a^2)^n}}~dx$} +$$\int{\frac{dx}{(x^2+a^2)^n}}= +\frac{x}{2(n-1)a^2(x^2+a^2)^{n-1}}+\frac{2n-3}{(2n-2)a^2} +\int{\frac{dx}{(x^2+a^2)^{n-1}}} +$$ +<<*>>= +)clear all + +--S 15 of 19 +aa:=integrate(1/((x^2+a^2)^n),x) +--R +--R +--R x +--R ++ 1 +--I (1) | ----------- d%L +--R ++ 2 2 n +--I (a + %L ) +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.140~~~~~$\displaystyle\int{\frac{x~dx}{(x^2+a^2)^n}}~dx$} +$$\int{\frac{x~dx}{(x^2+a^2)^n}}= +\frac{-1}{2(n-1)(x^2+a^2)^{n-1}} +$$ +<<*>>= +)clear all + +--S 16 of 19 +aa:=integrate(x/((x^2+a^2)^n),x) +--R +--R +--R 2 2 +--R - x - a +--R (1) ------------------------ +--R 2 2 +--R n log(x + a ) +--R (2n - 2)%e +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.141~~~~~$\displaystyle\int{\frac{dx}{x(x^2+a^2)^n}}~dx$} +$$\int{\frac{dx}{x(x^2+a^2)^n}}= +\frac{1}{2(n-1)a^2(x^2+a^2)^{n-1}}+\frac{1}{a^2} +\int{\frac{dx}{x(x^2+a^2)^{n-1}}} +$$ +<<*>>= +)clear all + +--S 17 of 19 +aa:=integrate(1/(x*(x^2+a^2)^n),x) +--R +--R +--R x +--R ++ 1 +--I (1) | -------------- d%L +--R ++ 2 2 n +--I %L (a + %L ) +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.142~~~~~$\displaystyle\int{\frac{x^mdx}{(x^2+a^2)^n}}~dx$} +$$\int{\frac{x^mdx}{(x^2+a^2)^n}}= +\int{\frac{x^{m-2}dx}{(x^2+a^2)^{n-1}}} - +a^2\int{\frac{x^{m-2}dx}{(x^2+a^2)^n}} +$$ +<<*>>= +)clear all + +--S 18 of 19 +aa:=integrate(x^m/((x^2+a^2)^n),x) +--R +--R +--R x m +--I ++ %L +--I (1) | ----------- d%L +--R ++ 2 2 n +--I (a + %L ) +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.143~~~~~$\displaystyle\int{\frac{dx}{x^m(x^2+a^2)^n}}~dx$} +$$\int{\frac{dx}{x^m(x^2+a^2)^n}}= +\frac{1}{a^2}\int{\frac{dx}{x^m(x^2+a^2)^{n-1}}}- +\frac{1}{a^2}\int{\frac{dx}{x^{m-2}(x^2+a^2)^n}} +$$ +<<*>>= +)clear all + +--S 19 of 19 +aa:=integrate(1/(x^m*(x^2+a^2)^n),x) +--R +--R +--R x +--R ++ 1 +--I (1) | -------------- d%L +--R ++ m 2 2 n +--I %L (a + %L ) +--R Type: Union(Expression Integer,...) +--E + +)spool +)lisp (bye) +@ + +\eject +\begin{thebibliography}{99} +\bibitem{1} Spiegel, Murray R. +{\sl Mathematical Handbook of Formulas and Tables}\\ +Schaum's Outline Series McGraw-Hill 1968 p64 +\end{thebibliography} +\end{document}