diff --git a/changelog b/changelog index 456e2f9..502aa1e 100644 --- a/changelog +++ b/changelog @@ -1,3 +1,6 @@ +20080417 tpd src/input/schaum10.input show Schaums-Axiom equivalence +20080417 tpd src/input/schaum9.input show Schaums-Axiom equivalence +20080417 tpd src/input/schaum8.input show Schaums-Axiom equivalence 20080417 tpd src/input/schaum7.input fix typo in --S 30 20080417 tpd src/input/schaum5.input fix typo 14:88 -> 14:123 20080416 tpd src/input/schaum7.input show Schaums-Axiom equivalence diff --git a/src/input/schaum10.input.pamphlet b/src/input/schaum10.input.pamphlet index 8582086..d0ba7d9 100644 --- a/src/input/schaum10.input.pamphlet +++ b/src/input/schaum10.input.pamphlet @@ -15,7 +15,7 @@ $$\int{\frac{1}{\sqrt{x^2-a^2}}}=\ln\left(x+\sqrt{x^2-a^2}\right)$$ )set message auto off )clear all ---S 1 of 28 +--S 1 aa:=integrate(1/(sqrt(x^2-a^2)),x) --R --R @@ -24,6 +24,39 @@ aa:=integrate(1/(sqrt(x^2-a^2)),x) --R (1) - log(\|x - a - x) --R Type: Union(Expression Integer,...) --E + +--S 2 +bb:=log(x+sqrt(x^2-a^2)) +--R +--R +-------+ +--R | 2 2 +--R (2) log(\|x - a + x) +--R Type: Expression Integer +--E + +--S 3 +cc:=aa-bb +--R +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R (3) - log(\|x - a + x) - log(\|x - a - x) +--R Type: Expression Integer +--E + +--S 4 +logmul1:=rule(c*log(a)+c*log(b) == c*log(a*b)) +--R +--I (4) c log(b) + c log(a) + %I == c log(a b) + %I +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 5 14:210 Schaums and Axiom differ by a constant +dd:=logmul1 cc +--R +--R 2 +--R (5) - log(- a ) +--R Type: Expression Integer +--E @ \section{\cite{1}:14.xxx~~~~~$\displaystyle\int{\frac{x~dx}{\sqrt{x^2-a^2}}}$} @@ -31,7 +64,7 @@ $$\int{\frac{x}{\sqrt{x^2-a^2}}}=\sqrt{x^2-a^2}$$ <<*>>= )clear all ---S 2 of 28 +--S 6 aa:=integrate(x/(sqrt(x^2-a^2)),x) --R --R @@ -44,6 +77,22 @@ aa:=integrate(x/(sqrt(x^2-a^2)),x) --R \|x - a - x --R Type: Union(Expression Integer,...) --E + +--S 7 +bb:=sqrt(x^2-a^2) +--R +--R +-------+ +--R | 2 2 +--R (2) \|x - a +--R Type: Expression Integer +--E + +--S 8 14:xxx Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.211~~~~~$\displaystyle @@ -54,7 +103,7 @@ $$ <<*>>= )clear all ---S 3 of 28 +--S 9 aa:=integrate(x^2/sqrt(x^2-a^2),x) --R --R @@ -72,6 +121,29 @@ aa:=integrate(x^2/sqrt(x^2-a^2),x) --R 4x\|x - a - 4x + 2a --R Type: Union(Expression Integer,...) --E + +--S 10 +bb:=(x*sqrt(x^2-a^2))/2+a^2/2*log(x+sqrt(x^2-a^2)) +--R +--R +-------+ +-------+ +--R 2 | 2 2 | 2 2 +--R a log(\|x - a + x) + x\|x - a +--R (2) ----------------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 11 14:211 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R +-------+ +-------+ +--R 2 | 2 2 2 | 2 2 +--R - a log(\|x - a + x) - a log(\|x - a - x) +--R (3) ----------------------------------------------- +--R 2 +--R Type: Expression Integer +--E + @ \section{\cite{1}:14.212~~~~~$\displaystyle @@ -82,7 +154,7 @@ $$ <<*>>= )clear all ---S 4 of 28 +--S 12 aa:=integrate(x^3/sqrt(x^2-a^2),x) --R --R @@ -95,6 +167,24 @@ aa:=integrate(x^3/sqrt(x^2-a^2),x) --R (12x - 3a )\|x - a - 12x + 9a x --R Type: Union(Expression Integer,...) --E + +--S 13 +bb:=(x^2-a^2)^(3/2)/3+a^2*sqrt(x^2-a^2) +--R +--R +-------+ +--R 2 2 | 2 2 +--R (x + 2a )\|x - a +--R (2) -------------------- +--R 3 +--R Type: Expression Integer +--E + +--S 14 14:212 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.213~~~~~$\displaystyle\int{\frac{dx}{x\sqrt{x^2-a^2}}}$} @@ -104,7 +194,7 @@ $$ <<*>>= )clear all ---S 5 of 28 +--S 15 aa:=integrate(1/(x*sqrt(x^2-a^2)),x) --R --R @@ -117,6 +207,30 @@ aa:=integrate(1/(x*sqrt(x^2-a^2)),x) --R a --R Type: Union(Expression Integer,...) --E + +--S 16 +bb:=1/a*asec(abs(x/a)) +--R +--R x +--R asec(abs(-)) +--R a +--R (2) ------------ +--R a +--R Type: Expression Integer +--E + +--S 17 14:213 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R +-------+ +--R | 2 2 +--R \|x - a - x x +--R 2atan(--------------) - asec(abs(-)) +--R a a +--R (3) ------------------------------------ +--R a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.214~~~~~$\displaystyle @@ -127,7 +241,7 @@ $$ <<*>>= )clear all ---S 6 of 28 +--S 18 aa:=integrate(1/(x^2*sqrt(x^2-a^2)),x) --R --R @@ -138,6 +252,28 @@ aa:=integrate(1/(x^2*sqrt(x^2-a^2)),x) --R x\|x - a - x --R Type: Union(Expression Integer,...) --E + +--S 19 +bb:=sqrt(x^2-a^2)/(a^2*x) +--R +--R +-------+ +--R | 2 2 +--R \|x - a +--R (2) ---------- +--R 2 +--R a x +--R Type: Expression Integer +--E + +--S 20 14:214 Schaums and Axiom differ by a constant +cc:=aa-bb +--R +--R 1 +--R (3) -- +--R 2 +--R a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.215~~~~~$\displaystyle\int{\frac{dx}{x^3\sqrt{x^2-a^2}}}$} @@ -148,7 +284,7 @@ $$ <<*>>= )clear all ---S 7 of 28 +--S 21 aa:=integrate(1/(x^3*sqrt(x^2-a^2)),x) --R --R @@ -168,6 +304,34 @@ aa:=integrate(1/(x^3*sqrt(x^2-a^2)),x) --R 4a x \|x - a - 4a x + 2a x --R Type: Union(Expression Integer,...) --E + +--S 22 +bb:=sqrt(x^2-a^2)/(2*a^2*x^2)+1/(2*a^3)*asec(abs(x/a)) +--R +--R +-------+ +--R 2 x | 2 2 +--R x asec(abs(-)) + a\|x - a +--R a +--R (2) ---------------------------- +--R 3 2 +--R 2a x +--R Type: Expression Integer +--E + +--S 23 14:215 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R +-------+ +--R | 2 2 +--R \|x - a - x x +--R 2atan(--------------) - asec(abs(-)) +--R a a +--R (3) ------------------------------------ +--R 3 +--R 2a +--R Type: Expression Integer +--E + @ \section{\cite{1}:14.216~~~~~$\displaystyle\int{\sqrt{x^2-a^2}}~dx$} @@ -177,7 +341,7 @@ $$ <<*>>= )clear all ---S 8 of 28 +--S 24 aa:=integrate(sqrt(x^2-a^2),x) --R --R @@ -195,6 +359,29 @@ aa:=integrate(sqrt(x^2-a^2),x) --R 4x\|x - a - 4x + 2a --R Type: Union(Expression Integer,...) --E + +--S 25 +bb:=(x*sqrt(x^2-a^2))/2-a^2/2*log(x+sqrt(x^2-a^2)) +--R +--R +-------+ +-------+ +--R 2 | 2 2 | 2 2 +--R - a log(\|x - a + x) + x\|x - a +--R (2) ------------------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 26 14:216 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R +-------+ +-------+ +--R 2 | 2 2 2 | 2 2 +--R a log(\|x - a + x) + a log(\|x - a - x) +--R (3) --------------------------------------------- +--R 2 +--R Type: Expression Integer +--E + @ \section{\cite{1}:14.217~~~~~$\displaystyle\int{x\sqrt{x^2-a^2}}~dx$} @@ -204,7 +391,7 @@ $$ <<*>>= )clear all ---S 9 of 28 +--S 27 aa:=integrate(x*sqrt(x^2-a^2),x) --R --R @@ -217,6 +404,24 @@ aa:=integrate(x*sqrt(x^2-a^2),x) --R (12x - 3a )\|x - a - 12x + 9a x --R Type: Union(Expression Integer,...) --E + +--S 28 +bb:=(x^2-a^2)^(3/2)/3 +--R +--R +-------+ +--R 2 2 | 2 2 +--R (x - a )\|x - a +--R (2) ------------------- +--R 3 +--R Type: Expression Integer +--E + +--S 29 14:217 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.218~~~~~$\displaystyle @@ -228,7 +433,7 @@ $$ <<*>>= )clear all ---S 10 of 28 +--S 30 aa:=integrate(x^2*sqrt(x^2-a^2),x) --R --R @@ -246,6 +451,28 @@ aa:=integrate(x^2*sqrt(x^2-a^2),x) --R (64x - 32a x)\|x - a - 64x + 64a x - 8a --R Type: Union(Expression Integer,...) --E + +--S 31 +bb:=(x*(x^2-a^2)^(3/2))/4+(a^2*x*sqrt(x^2-a^2))/8-a^4/8*log(x+sqrt(x^2-a^2)) +--R +--R +-------+ +-------+ +--R 4 | 2 2 3 2 | 2 2 +--R - a log(\|x - a + x) + (2x - a x)\|x - a +--R (2) ----------------------------------------------- +--R 8 +--R Type: Expression Integer +--E + +--S 32 14:218 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R +-------+ +-------+ +--R 4 | 2 2 4 | 2 2 +--R a log(\|x - a + x) + a log(\|x - a - x) +--R (3) --------------------------------------------- +--R 8 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.219~~~~~$\displaystyle @@ -256,7 +483,7 @@ $$ <<*>>= )clear all ---S 11 of 28 +--S 33 aa:=integrate(x^3*sqrt(x^2-a^2),x) --R --R @@ -273,6 +500,24 @@ aa:=integrate(x^3*sqrt(x^2-a^2),x) --R (240x - 180a x + 15a )\|x - a - 240x + 300a x - 75a x --R Type: Union(Expression Integer,...) --E + +--S 34 +bb:=(x^2-a^2)^(5/2)/5+(a^2*(x^2-a^2)^(3/2))/3 +--R +--R +-------+ +--R 4 2 2 4 | 2 2 +--R (3x - a x - 2a )\|x - a +--R (2) ---------------------------- +--R 15 +--R Type: Expression Integer +--E + +--S 35 14:219 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.220~~~~~$\displaystyle @@ -283,7 +528,7 @@ $$ <<*>>= )clear all ---S 12 of 28 +--S 36 aa:=integrate(sqrt(x^2-a^2)/x,x) --R --R @@ -298,6 +543,27 @@ aa:=integrate(sqrt(x^2-a^2)/x,x) --R \|x - a - x --R Type: Union(Expression Integer,...) --E + +--S 37 +bb:=sqrt(x^2-a^2)-a*asec(abs(x/a)) +--R +--R +-------+ +--R x | 2 2 +--R (2) - a asec(abs(-)) + \|x - a +--R a +--R Type: Expression Integer +--E + +--S 38 14:220 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R +-------+ +--R | 2 2 +--R \|x - a - x x +--R (3) - 2a atan(--------------) + a asec(abs(-)) +--R a a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.221~~~~~$\displaystyle @@ -308,7 +574,7 @@ $$ <<*>>= )clear all ---S 13 of 28 +--S 39 aa:=integrate(sqrt(x^2-a^2)/x^2,x) --R --R @@ -321,6 +587,27 @@ aa:=integrate(sqrt(x^2-a^2)/x^2,x) --R x\|x - a - x --R Type: Union(Expression Integer,...) --E + +--S 40 +bb:=-sqrt(x^2-a^2)/x+log(x+sqrt(x^2-a^2)) +--R +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R x log(\|x - a + x) - \|x - a +--R (2) ---------------------------------- +--R x +--R Type: Expression Integer +--E + +--S 41 14:221 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R (3) - log(\|x - a + x) - log(\|x - a - x) - 1 +--R Type: Expression Integer +--E + @ \section{\cite{1}:14.222~~~~~$\displaystyle @@ -332,7 +619,7 @@ $$ <<*>>= )clear all ---S 14 of 28 +--S 42 aa:=integrate(sqrt(x^2-a^2)/x^3,x) --R --R @@ -352,6 +639,32 @@ aa:=integrate(sqrt(x^2-a^2)/x^3,x) --R 4a x \|x - a - 4a x + 2a x --R Type: Union(Expression Integer,...) --E + +--S 43 +bb:=-sqrt(x^2-a^2)/(2*x^2)+1/(2*a)*asec(abs(x/a)) +--R +--R +-------+ +--R 2 x | 2 2 +--R x asec(abs(-)) - a\|x - a +--R a +--R (2) ---------------------------- +--R 2 +--R 2a x +--R Type: Expression Integer +--E + +--S 44 14:222 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R +-------+ +--R | 2 2 +--R \|x - a - x x +--R 2atan(--------------) - asec(abs(-)) +--R a a +--R (3) ------------------------------------ +--R 2a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.223~~~~~$\displaystyle\int{\frac{dx}{(x^2-a^2)^{3/2}}}$} @@ -361,7 +674,7 @@ $$ <<*>>= )clear all ---S 15 of 28 +--S 45 aa:=integrate(1/(x^2-a^2)^(3/2),x) --R --R @@ -372,6 +685,27 @@ aa:=integrate(1/(x^2-a^2)^(3/2),x) --R x\|x - a - x + a --R Type: Union(Expression Integer,...) --E + +--S 46 +bb:=-x/(a^2*sqrt(x^2-a^2)) +--R +--R x +--R (2) - ------------ +--R +-------+ +--R 2 | 2 2 +--R a \|x - a +--R Type: Expression Integer +--E + +--S 47 14:223 Schaums and Axiom differ by a constant +cc:=aa-bb +--R +--R 1 +--R (3) - -- +--R 2 +--R a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.224~~~~~$\displaystyle @@ -382,7 +716,7 @@ $$ <<*>>= )clear all ---S 16 of 28 +--S 48 aa:=integrate(x/(x^2-a^2)^(3/2),x) --R --R @@ -395,6 +729,24 @@ aa:=integrate(x/(x^2-a^2)^(3/2),x) --R x\|x - a - x + a --R Type: Union(Expression Integer,...) --E + +--S 49 +bb:=-1/sqrt(x^2-a^2) +--R +--R 1 +--R (2) - ---------- +--R +-------+ +--R | 2 2 +--R \|x - a +--R Type: Expression Integer +--E + +--S 50 14:224 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.225~~~~~$\displaystyle @@ -405,7 +757,7 @@ $$ <<*>>= )clear all ---S 17 of 28 +--S 51 aa:=integrate(x^2/(x^2-a^2)^(3/2),x) --R --R @@ -418,6 +770,28 @@ aa:=integrate(x^2/(x^2-a^2)^(3/2),x) --R x\|x - a - x + a --R Type: Union(Expression Integer,...) --E + +--S 52 +bb:=-x/sqrt(x^2-a^2)+log(x+sqrt(x^2-a^2)) +--R +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R \|x - a log(\|x - a + x) - x +--R (2) --------------------------------- +--R +-------+ +--R | 2 2 +--R \|x - a +--R Type: Expression Integer +--E + +--S 53 14:225 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R (3) - log(\|x - a + x) - log(\|x - a - x) - 1 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.226~~~~~$\displaystyle @@ -428,7 +802,7 @@ $$ <<*>>= )clear all ---S 18 of 28 +--S 54 aa:=integrate(x^3/(x^2-a^2)^(3/2),x) --R --R @@ -441,6 +815,25 @@ aa:=integrate(x^3/(x^2-a^2)^(3/2),x) --R (2x - a )\|x - a - 2x + 2a x --R Type: Union(Expression Integer,...) --E + +--S 55 +bb:=sqrt(x^2-a^2)-a^2/sqrt(x^2-a^2) +--R +--R 2 2 +--R x - 2a +--R (2) ---------- +--R +-------+ +--R | 2 2 +--R \|x - a +--R Type: Expression Integer +--E + +--S 56 14:226 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.227~~~~~$\displaystyle @@ -452,7 +845,7 @@ $$ <<*>>= )clear all ---S 19 of 28 +--S 57 aa:=integrate(1/(x*(x^2-a^2)^(3/2)),x) --R --R @@ -467,6 +860,34 @@ aa:=integrate(1/(x*(x^2-a^2)^(3/2)),x) --R a x\|x - a - a x + a --R Type: Union(Expression Integer,...) --E + +--S 58 +bb:=-1/(a^2*sqrt(x^2-a^2))-1/a^3*asec(abs(x/a)) +--R +--R +-------+ +--R | 2 2 x +--R - \|x - a asec(abs(-)) - a +--R a +--R (2) ---------------------------- +--R +-------+ +--R 3 | 2 2 +--R a \|x - a +--R Type: Expression Integer +--E + +--S 59 14:227 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R +-------+ +--R | 2 2 +--R \|x - a - x x +--R - 2atan(--------------) + asec(abs(-)) +--R a a +--R (3) -------------------------------------- +--R 3 +--R a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.228~~~~~$\displaystyle @@ -477,7 +898,7 @@ $$ <<*>>= )clear all ---S 20 of 28 +--S 60 aa:=integrate(1/(x^2*(x^2-a^2)^(3/2)),x) --R --R @@ -488,6 +909,29 @@ aa:=integrate(1/(x^2*(x^2-a^2)^(3/2)),x) --R (2x - a x)\|x - a - 2x + 2a x --R Type: Union(Expression Integer,...) --E + +--S 61 +bb:=-sqrt(x^2-a^2)/(a^4*x)-x/(a^4*sqrt(x^2-a^2)) +--R +--R 2 2 +--R - 2x + a +--R (2) ------------- +--R +-------+ +--R 4 | 2 2 +--R a x\|x - a +--R Type: Expression Integer +--E + +--S 62 14:228 Schaums and Axiom differ by a constant +cc:=aa-bb +--R +--R 2 +--R (3) - -- +--R 4 +--R a +--R Type: Expression Integer +--E + @ \section{\cite{1}:14.229~~~~~$\displaystyle @@ -500,7 +944,7 @@ $$ <<*>>= )clear all ---S 21 of 28 +--S 63 aa:=integrate(1/(x^3*(x^2-a^2)^(3/2)),x) --R --R @@ -520,6 +964,34 @@ aa:=integrate(1/(x^3*(x^2-a^2)^(3/2)),x) --R (8a x - 6a x )\|x - a - 8a x + 10a x - 2a x --R Type: Union(Expression Integer,...) --E + +--S 64 +bb:=1/(2*a^2*x^2*sqrt(x^2-a^2))-3/(2*a^4*sqrt(x^2-a^2))-3/(2*a^5)*asec(abs(x/a)) +--R +--R +-------+ +--R 2 | 2 2 x 2 3 +--R - 3x \|x - a asec(abs(-)) - 3a x + a +--R a +--R (2) ---------------------------------------- +--R +-------+ +--R 5 2 | 2 2 +--R 2a x \|x - a +--R Type: Expression Integer +--E + +--S 65 14:229 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R +-------+ +--R | 2 2 +--R \|x - a - x x +--R - 6atan(--------------) + 3asec(abs(-)) +--R a a +--R (3) --------------------------------------- +--R 5 +--R 2a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.230~~~~~$\displaystyle\int{(x^2-a^2)^{3/2}}~dx$} @@ -530,7 +1002,7 @@ $$ <<*>>= )clear all ---S 22 of 28 +--S 66 aa:=integrate((x^2-a^2)^(3/2),x) --R --R @@ -551,6 +1023,29 @@ aa:=integrate((x^2-a^2)^(3/2),x) --R (64x - 32a x)\|x - a - 64x + 64a x - 8a --R Type: Union(Expression Integer,...) --E + +--S 67 +bb:=(x*(x^2-a^2)^(3/2))/4-(3*a^2*x*sqrt(x^2-a^2))/8+3/8*a^4*log(x+sqrt(x^2-a^2)) +--R +--R +-------+ +-------+ +--R 4 | 2 2 3 2 | 2 2 +--R 3a log(\|x - a + x) + (2x - 5a x)\|x - a +--R (2) ----------------------------------------------- +--R 8 +--R Type: Expression Integer +--E + +--S 68 14:230 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R +-------+ +-------+ +--R 4 | 2 2 4 | 2 2 +--R - 3a log(\|x - a + x) - 3a log(\|x - a - x) +--R (3) ------------------------------------------------- +--R 8 +--R Type: Expression Integer +--E + @ \section{\cite{1}:14.231~~~~~$\displaystyle\int{x(x^2-a^2)^{3/2}}~dx$} @@ -558,7 +1053,7 @@ $$\int{x(x^2-a^2)^{3/2}}=\frac{(x^2-a^2)^{5/2}}{5}$$ <<*>>= )clear all ---S 23 of 28 +--S 69 aa:=integrate(x*(x^2-a^2)^(3/2),x) --R --R @@ -575,6 +1070,24 @@ aa:=integrate(x*(x^2-a^2)^(3/2),x) --R (80x - 60a x + 5a )\|x - a - 80x + 100a x - 25a x --R Type: Union(Expression Integer,...) --E + +--S 70 +bb:=(x^2-a^2)^(5/2)/5 +--R +--R +-------+ +--R 4 2 2 4 | 2 2 +--R (x - 2a x + a )\|x - a +--R (2) --------------------------- +--R 5 +--R Type: Expression Integer +--E + +--S 71 14:231 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.232~~~~~$\displaystyle\int{x^2(x^2-a^2)^{3/2}}~dx$} @@ -586,7 +1099,7 @@ $$ <<*>>= )clear all ---S 24 of 28 +--S 72 aa:=integrate(x^2*(x^2-a^2)^(3/2),x) --R --R @@ -614,6 +1127,28 @@ aa:=integrate(x^2*(x^2-a^2)^(3/2),x) --R (1536x - 1536a x + 288a x)\|x - a - 1536x + 2304a x - 864a x + 48a --R Type: Union(Expression Integer,...) --E + +--S 73 +bb:=(x*(x^2-a^2)^(5/2))/6+(a^2*x*(x^2-a^2)^(3/2))/24-(a^4*x*sqrt(x^2-a^2))/16+a^6/16*log(x+sqrt(x^2-a^2)) +--R +--R +-------+ +-------+ +--R 6 | 2 2 5 2 3 4 | 2 2 +--R 3a log(\|x - a + x) + (8x - 14a x + 3a x)\|x - a +--R (2) -------------------------------------------------------- +--R 48 +--R Type: Expression Integer +--E + +--S 74 14:232 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R +-------+ +-------+ +--R 6 | 2 2 6 | 2 2 +--R - a log(\|x - a + x) - a log(\|x - a - x) +--R (3) ----------------------------------------------- +--R 16 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.233~~~~~$\displaystyle\int{x^3(x^2-a^2)^{3/2}}~dx$} @@ -623,7 +1158,7 @@ $$ <<*>>= )clear all ---S 25 of 28 +--S 75 aa:=integrate(x^3*(x^2-a^2)^(3/2),x) --R --R @@ -652,6 +1187,24 @@ aa:=integrate(x^3*(x^2-a^2)^(3/2),x) --R - 1960a x + 245a x --R Type: Union(Expression Integer,...) --E + +--S 76 +bb:=(x^2-a^2)^(7/2)/7+(a^2*(x^2-a^2)^(5/2))/5 +--R +--R +-------+ +--R 6 2 4 4 2 6 | 2 2 +--R (5x - 8a x + a x + 2a )\|x - a +--R (2) ------------------------------------ +--R 35 +--R Type: Expression Integer +--E + +--S 77 14:233 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.234~~~~~$\displaystyle @@ -663,7 +1216,7 @@ $$ <<*>>= )clear all ---S 26 of 28 +--S 78 aa:=integrate((x^2-a^2)^(3/2)/x,x) --R --R @@ -683,6 +1236,29 @@ aa:=integrate((x^2-a^2)^(3/2)/x,x) --R (12x - 3a )\|x - a - 12x + 9a x --R Type: Union(Expression Integer,...) --E + +--S 79 +bb:=(x^2-a^2)^(3/2)/3-a^2*sqrt(x^2-a^2)+a^3*asec(abs(x/a)) +--R +--R +-------+ +--R 3 x 2 2 | 2 2 +--R 3a asec(abs(-)) + (x - 4a )\|x - a +--R a +--R (2) -------------------------------------- +--R 3 +--R Type: Expression Integer +--E + +--S 80 14:234 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R +-------+ +--R | 2 2 +--R 3 \|x - a - x 3 x +--R (3) 2a atan(--------------) - a asec(abs(-)) +--R a a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.235~~~~~$\displaystyle @@ -694,7 +1270,7 @@ $$ <<*>>= )clear all ---S 27 of 28 +--S 81 aa:=integrate((x^2-a^2)^{3/2}/x^2,x) --R --R @@ -712,6 +1288,28 @@ aa:=integrate((x^2-a^2)^{3/2}/x^2,x) --R (8x - 2a x)\|x - a - 8x + 6a x --R Type: Union(Expression Integer,...) --E + +--S 82 +bb:=-(x^2-a^2)^(3/2)/x+3*x*sqrt(x^2-a^2)/2-3/2*a^2*log(x+sqrt(x^2-a^2)) +--R +--R +-------+ +-------+ +--R 2 | 2 2 2 2 | 2 2 +--R - 3a x log(\|x - a + x) + (x + 2a )\|x - a +--R (2) ------------------------------------------------- +--R 2x +--R Type: Expression Integer +--E + +--S 83 14:235 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R +-------+ +-------+ +--R 2 | 2 2 2 | 2 2 2 +--R 3a log(\|x - a + x) + 3a log(\|x - a - x) + 2a +--R (3) ----------------------------------------------------- +--R 2 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.236~~~~~$\displaystyle @@ -723,7 +1321,7 @@ $$ <<*>>= )clear all ---S 28 of 28 +--S 84 aa:=integrate((x^2-a^2)^(3/2)/x^3,x) --R --R @@ -744,6 +1342,31 @@ aa:=integrate((x^2-a^2)^(3/2)/x^3,x) --R Type: Union(Expression Integer,...) --E +--S 85 +bb:=-(x^2-a^2)^(3/2)/(2*x^2)+(3*sqrt(x^2-a^2))/2-3/2*a*asec(abs(x/a)) +--R +--R +-------+ +--R 2 x 2 2 | 2 2 +--R - 3a x asec(abs(-)) + (2x + a )\|x - a +--R a +--R (2) ------------------------------------------ +--R 2 +--R 2x +--R Type: Expression Integer +--E + +--S 86 14:236 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R +-------+ +--R | 2 2 +--R \|x - a - x x +--R - 6a atan(--------------) + 3a asec(abs(-)) +--R a a +--R (3) ------------------------------------------- +--R 2 +--R Type: Expression Integer +--E )spool )lisp (bye) @ diff --git a/src/input/schaum8.input.pamphlet b/src/input/schaum8.input.pamphlet index a5c0c7e..778d135 100644 --- a/src/input/schaum8.input.pamphlet +++ b/src/input/schaum8.input.pamphlet @@ -16,7 +16,7 @@ $$\int{\frac{1}{a^2-x^2}}=-\frac{1}{a}\coth^{-1}\frac{x}{a}$$ )set message auto off )clear all ---S 1 of 19 +--S 1 aa:=integrate(1/(a^2-x^2),x) --R --R @@ -25,6 +25,63 @@ aa:=integrate(1/(a^2-x^2),x) --R 2a --R Type: Union(Expression Integer,...) --E + +--S 2 +bb:=1/(2*a)*log((a+x)/(a-x)) +--R +--R - x - a +--R log(-------) +--R x - a +--R (2) ------------ +--R 2a +--R Type: Expression Integer +--E + +--S 3 +cc:=aa-bb +--R +--R - x - a +--R log(x + a) - log(x - a) - log(-------) +--R x - a +--R (3) -------------------------------------- +--R 2a +--R Type: Expression Integer +--E + +--S 4 +divlog:=rule(log(a/b) == log(a) - log(b)) +--R +--R a +--R (4) log(-) == - log(b) + log(a) +--R b +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 5 +dd:=divlog cc +--R +--R log(x + a) - log(- x - a) +--R (5) ------------------------- +--R 2a +--R Type: Expression Integer +--E + +--S 6 +logminus:=rule(log(x + a) - log(- x - a) == log(-1)) +--R +--I (6) log(x + a) - log(- x - a) + %I == log(- 1) + %I +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 7 14:163 Schaums and Axiom differ by a constant +ee:=logminus dd +--R +--R log(- 1) +--R (7) -------- +--R 2a +--R Type: Expression Integer +--E + @ \section{\cite{1}:14.164~~~~~$\displaystyle\int{\frac{x~dx}{a^2-x^2}}$} @@ -32,7 +89,7 @@ $$\int{\frac{x}{a^2-x^2}}=-\frac{1}{2}\ln(a^2-x^2)$$ <<*>>= )clear all ---S 2 of 19 +--S 8 aa:=integrate(x/(a^2-x^2),x) --R --R @@ -42,6 +99,43 @@ aa:=integrate(x/(a^2-x^2),x) --R 2 --R Type: Union(Expression Integer,...) --E + +--S 9 +bb:=-1/2*log(a^2-x^2) +--R +--R 2 2 +--R log(- x + a ) +--R (2) - -------------- +--R 2 +--R Type: Expression Integer +--E + +--S 10 +cc:=aa-bb +--R +--R 2 2 2 2 +--R - log(x - a ) + log(- x + a ) +--R (3) ------------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 11 +logminus1:=rule(-log(x^2-a^2)+log(-x^2+a^2) == log(-1)) +--R +--R 2 2 2 2 +--I (4) - log(x - a ) + log(- x + a ) + %H == log(- 1) + %H +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 12 14:164 Schaums and Axiom differ by a constant +dd:=logminus1 cc +--R +--R log(- 1) +--R (5) -------- +--R 2 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.165~~~~~$\displaystyle\int{\frac{x^2~dx}{a^2-x^2}}$} @@ -49,7 +143,7 @@ $$\int{\frac{x^2}{a^2-x^2}}=-x+\frac{a}{2}\ln\left(\frac{a+x}{a-x}\right)$$ <<*>>= )clear all ---S 3 of 19 +--S 13 aa:=integrate(x^2/(a^2-x^2),x) --R --R @@ -58,6 +152,62 @@ aa:=integrate(x^2/(a^2-x^2),x) --R 2 --R Type: Union(Expression Integer,...) --E + +--S 14 +bb:=-x+a/2*log((a+x)/(a-x)) +--R +--R - x - a +--R a log(-------) - 2x +--R x - a +--R (2) ------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 15 +cc:=aa-bb +--R +--R - x - a +--R a log(x + a) - a log(x - a) - a log(-------) +--R x - a +--R (3) -------------------------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 16 +divlog:=rule(log(a/b) == log(a) - log(b)) +--R +--R a +--R (4) log(-) == - log(b) + log(a) +--R b +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 17 +dd:=divlog cc +--R +--R a log(x + a) - a log(- x - a) +--R (5) ----------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 18 +logminusa:=rule(b*log(x + a) - b*log(- x - a) == b*log(-1)) +--R +--I (6) b log(x + a) - b log(- x - a) + %M == b log(- 1) + %M +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 19 14:165 Schaums and Axiom differ by a constant +ee:=logminusa dd +--R +--R a log(- 1) +--R (7) ---------- +--R 2 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.166~~~~~$\displaystyle\int{\frac{x^3~dx}{a^2-x^2}}$} @@ -66,7 +216,7 @@ $$\int{\frac{x^3}{a^2-x^2}}=-\frac{x^2}{2}-\frac{a^2}{2}\ln(a^2-x^2)$$ <<*>>= )clear all ---S 4 of 19 +--S 20 aa:=integrate(x^3/(a^2-x^2),x) --R --R @@ -76,6 +226,45 @@ aa:=integrate(x^3/(a^2-x^2),x) --R 2 --R Type: Union(Expression Integer,...) --E + +--S 21 +bb:=-x^2/2-a^2/2*log(a^2-x^2) +--R +--R 2 2 2 2 +--R - a log(- x + a ) - x +--R (2) ----------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 22 +cc:=aa-bb +--R +--R 2 2 2 2 2 2 +--R - a log(x - a ) + a log(- x + a ) +--R (3) ----------------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 23 +logminus1b:=rule(-b*log(x^2-a^2)+b*log(-x^2+a^2) == b*log(-1)) +--R +--R 2 2 2 2 +--I (4) - b log(x - a ) + b log(- x + a ) + %N == b log(- 1) + %N +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 24 14:166 Schaums and Axiom differ by a constant +dd:=logminus1b cc +--R +--R 2 +--R a log(- 1) +--R (5) ---------- +--R 2 +--R Type: Expression Integer +--E + @ \section{\cite{1}:14.167~~~~~$\displaystyle\int{\frac{dx}{x(a^2-x^2)}}$} @@ -85,7 +274,7 @@ $$ <<*>>= )clear all ---S 5 of 19 +--S 25 aa:=integrate(1/(x*(a^2-x^2)),x) --R --R @@ -96,6 +285,72 @@ aa:=integrate(1/(x*(a^2-x^2)),x) --R 2a --R Type: Union(Expression Integer,...) --E + +--S 26 +bb:=1/(2*a^2)*log(x^2/(a^2-x^2)) +--R +--R 2 +--R x +--R log(- -------) +--R 2 2 +--R x - a +--R (2) -------------- +--R 2 +--R 2a +--R Type: Expression Integer +--E + +--S 27 +cc:=aa-bb +--R +--R 2 +--R 2 2 x +--R - log(x - a ) + 2log(x) - log(- -------) +--R 2 2 +--R x - a +--R (3) ----------------------------------------- +--R 2 +--R 2a +--R Type: Expression Integer +--E + +--S 28 +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 + +--S 29 +dd:=divlog cc +--R +--R 2 +--R 2log(x) - log(- x ) +--R (5) ------------------- +--R 2 +--R 2a +--R Type: Expression Integer +--E + +--S 30 +logpowminus:=rule(log(-a^n) == n*log(a)+log(-1)) +--R +--R n +--R (6) log(- a ) == n log(a) + log(- 1) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 31 14:167 Schaums and Axiom differ by a constant +ee:=logpowminus dd +--R +--R log(- 1) +--R (7) - -------- +--R 2 +--R 2a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.168~~~~~$\displaystyle\int{\frac{dx}{x^2(a^2-x^2)}}$} @@ -105,7 +360,7 @@ $$ <<*>>= )clear all ---S 6 of 19 +--S 32 aa:=integrate(1/(x^2*(a^2-x^2)),x) --R --R @@ -115,6 +370,67 @@ aa:=integrate(1/(x^2*(a^2-x^2)),x) --R 2a x --R Type: Union(Expression Integer,...) --E + +--S 33 +bb:=-1/(a^2*x)+1/(2*a^3)*log((a+x)/(a-x)) +--R +--R - x - a +--R x log(-------) - 2a +--R x - a +--R (2) ------------------- +--R 3 +--R 2a x +--R Type: Expression Integer +--E + +--S 34 +cc:=aa-bb +--R +--R - x - a +--R log(x + a) - log(x - a) - log(-------) +--R x - a +--R (3) -------------------------------------- +--R 3 +--R 2a +--R Type: Expression Integer +--E + +--S 35 +divlog:=rule(log(a/b) == log(a) - log(b)) +--R +--R a +--R (4) log(-) == - log(b) + log(a) +--R b +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 36 +dd:=divlog cc +--R +--R log(x + a) - log(- x - a) +--R (5) ------------------------- +--R 3 +--R 2a +--R Type: Expression Integer +--E + +--S 37 +logminus:=rule(log(x + a) - log(- x - a) == log(-1)) +--R +--I (6) log(x + a) - log(- x - a) + %O == log(- 1) + %O +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 38 14:168 Schaums and Axiom differ by a constant +ee:=logminus dd +--R +--R log(- 1) +--R (7) -------- +--R 3 +--R 2a +--R Type: Expression Integer +--E + @ \section{\cite{1}:14.169~~~~~$\displaystyle\int{\frac{dx}{x^3(a^2-x^2)}}$} @@ -124,7 +440,7 @@ $$ <<*>>= )clear all ---S 7 of 19 +--S 39 aa:=integrate(1/(x^3*(a^2-x^2)),x) --R --R @@ -135,6 +451,70 @@ aa:=integrate(1/(x^3*(a^2-x^2)),x) --R 2a x --R Type: Union(Expression Integer,...) --E + +--S 40 +bb:=-1/(2*a^2*x^2)+1/(2*a^4)*log(x^2/(a^2-x^2)) +--R +--R 2 +--R 2 x 2 +--R x log(- -------) - a +--R 2 2 +--R x - a +--R (2) --------------------- +--R 4 2 +--R 2a x +--R Type: Expression Integer +--E + +--S 41 +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 + +--S 42 +bb1:=divlog bb +--R +--R 2 2 2 2 2 2 +--R - x log(x - a ) + x log(- x ) - a +--R (4) ----------------------------------- +--R 4 2 +--R 2a x +--R Type: Expression Integer +--E + +--S 43 +cc:=aa-bb1 +--R +--R 2 +--R 2log(x) - log(- x ) +--R (5) ------------------- +--R 4 +--R 2a +--R Type: Expression Integer +--E + +--S 44 +logminuspow:=rule(log(-x^n) == n*log(x)+log(-1)) +--R +--R n +--R (6) log(- x ) == n log(x) + log(- 1) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 45 14:169 Schaums and Axiom differ by a constant +dd:=logminuspow cc +--R +--R log(- 1) +--R (7) - -------- +--R 4 +--R 2a +--R Type: Expression Integer +--E + @ \section{\cite{1}:14.170~~~~~$\displaystyle\int{\frac{dx}{(a^2-x^2)^2}}$} @@ -144,7 +524,7 @@ $$ <<*>>= )clear all ---S 8 of 19 +--S 46 aa:=integrate(1/((a^2-x^2)^2),x) --R --R @@ -155,6 +535,66 @@ aa:=integrate(1/((a^2-x^2)^2),x) --R 4a x - 4a --R Type: Union(Expression Integer,...) --E + +--S 47 +bb:=x/(2*a^2*(a^2-x^2))+1/(4*a^3)*log((a+x)/(a-x)) +--R +--R 2 2 - x - a +--R (x - a )log(-------) - 2a x +--R x - a +--R (2) ---------------------------- +--R 3 2 5 +--R 4a x - 4a +--R Type: Expression Integer +--E + +--S 48 +divlog:=rule(log(a/b) == log(a) - log(b)) +--R +--R a +--R (3) log(-) == - log(b) + log(a) +--R b +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 49 +bb1:=divlog bb +--R +--R 2 2 2 2 +--R (- x + a )log(x - a) + (x - a )log(- x - a) - 2a x +--R (4) ---------------------------------------------------- +--R 3 2 5 +--R 4a x - 4a +--R Type: Expression Integer +--E + +--S 50 +cc:=aa-bb1 +--R +--R log(x + a) - log(- x - a) +--R (5) ------------------------- +--R 3 +--R 4a +--R Type: Expression Integer +--E + +--S 51 +logminus:=rule(log(x + a) - log(- x - a) == log(-1)) +--R +--I (6) log(x + a) - log(- x - a) + %P == log(- 1) + %P +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 52 14:170 Schaums and Axiom differ by a constant +dd:=logminus cc +--R +--R log(- 1) +--R (7) -------- +--R 3 +--R 4a +--R Type: Expression Integer +--E + @ \section{\cite{1}:14.171~~~~~$\displaystyle\int{\frac{x~dx}{(a^2-x^2)^2}}$} @@ -164,7 +604,7 @@ $$ <<*>>= )clear all ---S 9 of 19 +--S 53 aa:=integrate(x/((a^2-x^2)^2),x) --R --R @@ -174,6 +614,24 @@ aa:=integrate(x/((a^2-x^2)^2),x) --R 2x - 2a --R Type: Union(Expression Integer,...) --E + +--S 54 +bb:=1/(2*(a^2-x^2)) +--R +--R 1 +--R (2) - --------- +--R 2 2 +--R 2x - 2a +--R Type: Fraction Polynomial Integer +--E + +--S 55 14:171 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E + @ \section{\cite{1}:14.172~~~~~$\displaystyle\int{\frac{x^2dx}{(a^2-x^2)^2}}$} @@ -183,7 +641,7 @@ $$ <<*>>= )clear all ---S 10 of 19 +--S 56 aa:=integrate(x^2/((a^2-x^2)^2),x) --R --R @@ -194,6 +652,63 @@ aa:=integrate(x^2/((a^2-x^2)^2),x) --R 4a x - 4a --R Type: Union(Expression Integer,...) --E + +--S 57 +bb:=x/(2*(a^2-x^2))-1/(4*a)*log((a+x)/(a-x)) +--R +--R 2 2 - x - a +--R (- x + a )log(-------) - 2a x +--R x - a +--R (2) ------------------------------ +--R 2 3 +--R 4a x - 4a +--R Type: Expression Integer +--E + +--S 58 +divlog:=rule(log(a/b) == log(a) - log(b)) +--R +--R a +--R (3) log(-) == - log(b) + log(a) +--R b +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 59 +bb1:=divlog bb +--R +--R 2 2 2 2 +--R (x - a )log(x - a) + (- x + a )log(- x - a) - 2a x +--R (4) ---------------------------------------------------- +--R 2 3 +--R 4a x - 4a +--R Type: Expression Integer +--E + +--S 60 +cc:=aa-bb1 +--R +--R - log(x + a) + log(- x - a) +--R (5) --------------------------- +--R 4a +--R Type: Expression Integer +--E + +--S 61 +logminus2:=rule(-log(x + a) + log(- x - a) == log(-1)) +--R +--I (6) - log(x + a) + log(- x - a) + %S == log(- 1) + %S +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 62 14:172 Schaums and Axiom differ by a constant +dd:=logminus2 cc +--R +--R log(- 1) +--R (7) -------- +--R 4a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.173~~~~~$\displaystyle\int{\frac{x^3dx}{(a^2-x^2)^2}}$} @@ -203,7 +718,7 @@ $$ <<*>>= )clear all ---S 11 of 19 +--S 63 aa:=integrate(x^3/((a^2-x^2)^2),x) --R --R @@ -214,6 +729,45 @@ aa:=integrate(x^3/((a^2-x^2)^2),x) --R 2x - 2a --R Type: Union(Expression Integer,...) --E + +--S 64 +bb:=a^2/(2*(a^2-x^2))+1/2*log(a^2-x^2) +--R +--R 2 2 2 2 2 +--R (x - a )log(- x + a ) - a +--R (2) ---------------------------- +--R 2 2 +--R 2x - 2a +--R Type: Expression Integer +--E + +--S 65 +cc:=aa-bb +--R +--R 2 2 2 2 +--R log(x - a ) - log(- x + a ) +--R (3) ----------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 66 +logminus3:=rule(log(x^2-a^2)-log(-x^2+a^2) == log(-1)) +--R +--R 2 2 2 2 +--I (4) log(x - a ) - log(- x + a ) + %T == log(- 1) + %T +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 67 14:173 Schaums and Axiom differ by a constant +dd:=logminus3 cc +--R +--R log(- 1) +--R (5) -------- +--R 2 +--R Type: Expression Integer +--E + @ \section{\cite{1}:14.174~~~~~$\displaystyle\int{\frac{dx}{x(a^2-x^2)^2}}$} @@ -223,7 +777,7 @@ $$ <<*>>= )clear all ---S 12 of 19 +--S 68 aa:=integrate(1/(x*(a^2-x^2)^2),x) --R --R @@ -234,6 +788,70 @@ aa:=integrate(1/(x*(a^2-x^2)^2),x) --R 2a x - 2a --R Type: Union(Expression Integer,...) --E + +--S 69 +bb:=1/(2*a^2*(a^2-x^2))+1/(2*a^4)*log(x^2/(a^2-x^2)) +--R +--R 2 +--R 2 2 x 2 +--R (x - a )log(- -------) - a +--R 2 2 +--R x - a +--R (2) ---------------------------- +--R 4 2 6 +--R 2a x - 2a +--R Type: Expression Integer +--E + +--S 70 +divlog:=rule(log(a/b) == log(a) - log(b)) +--R +--R a +--R (3) log(-) == - log(b) + log(a) +--R b +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 71 +bb1:=divlog bb +--R +--R 2 2 2 2 2 2 2 2 +--R (- x + a )log(x - a ) + (x - a )log(- x ) - a +--R (4) ------------------------------------------------- +--R 4 2 6 +--R 2a x - 2a +--R Type: Expression Integer +--E + +--S 72 +cc:=aa-bb1 +--R +--R 2 +--R 2log(x) - log(- x ) +--R (5) ------------------- +--R 4 +--R 2a +--R Type: Expression Integer +--E + +--S 73 +logpowminus:=rule(log(-a^n) == n*log(a)+log(-1)) +--R +--R n +--R (6) log(- a ) == n log(a) + log(- 1) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 74 14:174 Schaums and Axiom differ by a constant +dd:=logpowminus cc +--R +--R log(- 1) +--R (7) - -------- +--R 4 +--R 2a +--R Type: Expression Integer +--E + @ \section{\cite{1}:14.175~~~~~$\displaystyle\int{\frac{dx}{x^2(a^2-x^2)^2}}$} @@ -244,17 +862,88 @@ $$ <<*>>= )clear all ---S 13 of 19 -aa:=integrate(1/((a^2-x^2)^2),x) ---R +--S 75 +aa:=integrate(1/(x^2*(a^2-x^2)^2),x) --R ---R 2 2 2 2 ---R (x - a )log(x + a) + (- x + a )log(x - a) - 2a x ---R (1) -------------------------------------------------- ---R 3 2 5 ---R 4a x - 4a +--R 3 2 3 2 2 3 +--R (3x - 3a x)log(x + a) + (- 3x + 3a x)log(x - a) - 6a x + 4a +--R (1) --------------------------------------------------------------- +--R 5 3 7 +--R 4a x - 4a x --R Type: Union(Expression Integer,...) --E + +--S 76 +bb:=-1/(a^4*x)+x/(2*a^4*(a^2-x^2))+3/(4*a^5)*log((a+x)/(a-x)) +--R +--R 3 2 - x - a 2 3 +--R (3x - 3a x)log(-------) - 6a x + 4a +--R x - a +--R (2) -------------------------------------- +--R 5 3 7 +--R 4a x - 4a x +--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 b +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 78 +bb1:=divlog bb +--R +--R 3 2 3 2 2 3 +--R (- 3x + 3a x)log(x - a) + (3x - 3a x)log(- x - a) - 6a x + 4a +--R (4) ----------------------------------------------------------------- +--R 5 3 7 +--R 4a x - 4a x +--R Type: Expression Integer +--E + +--S 79 +cc:=aa-bb +--R +--R - x - a +--R 3log(x + a) - 3log(x - a) - 3log(-------) +--R x - a +--R (5) ----------------------------------------- +--R 5 +--R 4a +--R Type: Expression Integer +--E + +--S 80 +dd:=divlog cc +--R +--R 3log(x + a) - 3log(- x - a) +--R (6) --------------------------- +--R 5 +--R 4a +--R Type: Expression Integer +--E + +--S 81 +logminusb:=rule(b*log(x + a) - b*log(- x - a) == b*log(-1)) +--R +--I (7) b log(x + a) - b log(- x - a) + %U == b log(- 1) + %U +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 82 14:175 Schaums and Axiom differ by a constant +ee:=logminusb dd +--R +--R 3log(- 1) +--R (8) --------- +--R 5 +--R 4a +--R Type: Expression Integer +--E + @ \section{\cite{1}:14.176~~~~~$\displaystyle\int{\frac{dx}{x^3(a^2-x^2)^2}}$} @@ -265,7 +954,7 @@ $$ <<*>>= )clear all ---S 14 of 19 +--S 83 aa:=integrate(1/(x^3*(a^2-x^2)^2),x) --R --R @@ -276,6 +965,70 @@ aa:=integrate(1/(x^3*(a^2-x^2)^2),x) --R 2a x - 2a x --R Type: Union(Expression Integer,...) --E + +--S 84 +bb:=-1/(2*a^4*x^2)+1/(2*a^4*(a^2-x^2))+1/a^6*log(x^2/(a^2-x^2)) +--R +--R 2 +--R 4 2 2 x 2 2 4 +--R (2x - 2a x )log(- -------) - 2a x + a +--R 2 2 +--R x - a +--R (2) ---------------------------------------- +--R 6 4 8 2 +--R 2a x - 2a x +--R Type: Expression Integer +--E + +--S 85 +divlog:=rule(log(a/b) == log(a) - log(b)) +--R +--R a +--R (3) log(-) == - log(b) + log(a) +--R b +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 86 +bb1:=divlog bb +--R +--R 4 2 2 2 2 4 2 2 2 2 2 4 +--R (- 2x + 2a x )log(x - a ) + (2x - 2a x )log(- x ) - 2a x + a +--R (4) ----------------------------------------------------------------- +--R 6 4 8 2 +--R 2a x - 2a x +--R Type: Expression Integer +--E + +--S 87 +cc:=aa-bb1 +--R +--R 2 +--R 2log(x) - log(- x ) +--R (5) ------------------- +--R 6 +--R a +--R Type: Expression Integer +--E + +--S 88 +logpowminus:=rule(log(-a^n) == n*log(a)+log(-1)) +--R +--R n +--R (6) log(- a ) == n log(a) + log(- 1) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 89 14:176 Schaums and Axiom differ by a constant +dd:=logpowminus cc +--R +--R log(- 1) +--R (7) - -------- +--R 6 +--R a +--R Type: Expression Integer +--E + @ \section{\cite{1}:14.177~~~~~$\displaystyle\int{\frac{dx}{(a^2-x^2)^n}}$} @@ -286,7 +1039,7 @@ $$ <<*>>= )clear all ---S 15 of 19 +--S 90 14:177 Axiom cannot do this integration aa:=integrate(1/((a^2-x^2)^n),x) --R --R @@ -306,7 +1059,7 @@ $$ <<*>>= )clear all ---S 16 of 19 +--S 91 aa:=integrate(x/((a^2-x^2)^n),x) --R --R @@ -318,6 +1071,49 @@ aa:=integrate(x/((a^2-x^2)^n),x) --R (2n - 2)%e --R Type: Union(Expression Integer,...) --E + +--S 92 +bb:=1/(2*(n-1)*(a^2-x^2)^(n-1)) +--R +--R 1 +--R (2) ------------------------ +--R 2 2 n - 1 +--R (2n - 2)(- x + a ) +--R Type: Expression Integer +--E + +--S 93 +cc:=aa-bb +--R +--R 2 2 +--R n log(- x + a ) 2 2 2 2 n - 1 +--R - %e + (- x + a )(- x + a ) +--R (3) -------------------------------------------------- +--R 2 2 +--R 2 2 n - 1 n log(- x + a ) +--R (2n - 2)(- x + a ) %e +--R Type: Expression Integer +--E + +--S 94 +explog:=rule(%e^(n*log(x)) == x^n) +--R +--R n log(x) n +--R (4) %e == x +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 95 14:178 Axiom cannot simplify this expression +dd:=explog cc +--R +--R 2 2 n 2 2 2 2 n - 1 +--R - (- x + a ) + (- x + a )(- x + a ) +--R (5) -------------------------------------------- +--R 2 2 n - 1 2 2 n +--R (2n - 2)(- x + a ) (- x + a ) +--R Type: Expression Integer +--E + @ \section{\cite{1}:14.179~~~~~$\displaystyle\int{\frac{dx}{x(a^2-x^2)^n}}$} @@ -328,7 +1124,7 @@ $$ <<*>>= )clear all ---S 17 of 19 +--S 96 14:179 Axiom cannot integrate this expression aa:=integrate(1/(x*(a^2-x^2)^n),x) --R --R @@ -349,7 +1145,7 @@ $$ <<*>>= )clear all ---S 18 of 19 +--S 97 14:180 Axiom cannot integrate this expression aa:=integrate(x^m/((a^2-x^2)^n),x) --R --R @@ -370,7 +1166,7 @@ $$ <<*>>= )clear all ---S 19 of 19 +--S 98 14:181 Axiom cannot integrate this expression aa:=integrate(1/(x^m*(a^2-x^2)^n),x) --R --R diff --git a/src/input/schaum9.input.pamphlet b/src/input/schaum9.input.pamphlet index 6610811..fceb094 100644 --- a/src/input/schaum9.input.pamphlet +++ b/src/input/schaum9.input.pamphlet @@ -16,7 +16,7 @@ $$\int{\frac{1}{\sqrt{x^2+a^2}}}=\sinh^{-1}\frac{x}{a}$$ )set message auto off )clear all ---S 1 of 28 +--S 1 aa:=integrate(1/(sqrt(x^2+a^2)),x) --R --R @@ -25,14 +25,33 @@ aa:=integrate(1/(sqrt(x^2+a^2)),x) --R (1) - log(\|x + a - x) --R Type: Union(Expression Integer,...) --E + +--S 2 +bb:=log(x+sqrt(x^2+a^2)) +--R +--R +-------+ +--R | 2 2 +--R (2) log(\|x + a + x) +--R Type: Expression Integer +--E + +--S 3 14:182 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R (3) - log(\|x + a + x) - log(\|x + a - x) +--R Type: Expression Integer +--E @ +This is equal to $-\log(a^2)$ but Axiom cannot prove it. \section{\cite{1}:14.183~~~~~$\displaystyle\int{\frac{x~dx}{\sqrt{x^2+a^2}}}$} $$\int{\frac{x}{\sqrt{x^2+a^2}}}=\sqrt{x^2+a^2}$$ <<*>>= )clear all ---S 2 of 28 +--S 4 aa:=integrate(x/(sqrt(x^2+a^2)),x) --R --R @@ -45,6 +64,22 @@ aa:=integrate(x/(sqrt(x^2+a^2)),x) --R \|x + a - x --R Type: Union(Expression Integer,...) --E + +--S 5 +bb:=sqrt(x^2+a^2) +--R +--R +-------+ +--R | 2 2 +--R (2) \|x + a +--R Type: Expression Integer +--E + +--S 6 14:183 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.184~~~~~$\displaystyle @@ -55,7 +90,7 @@ $$ <<*>>= )clear all ---S 3 of 28 +--S 7 aa:=integrate(x^2/sqrt(x^2+a^2),x) --R --R @@ -73,6 +108,46 @@ aa:=integrate(x^2/sqrt(x^2+a^2),x) --R 4x\|x + a - 4x - 2a --R Type: Union(Expression Integer,...) --E + +--S 8 +bb:=(x*sqrt(x^2+a^2))/2-a^2/2*log(x+sqrt(x^2+a^2)) +--R +--R +-------+ +-------+ +--R 2 | 2 2 | 2 2 +--R - a log(\|x + a + x) + x\|x + a +--R (2) ------------------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 9 +cc:=aa-bb +--R +--R +-------+ +-------+ +--R 2 | 2 2 2 | 2 2 +--R a log(\|x + a + x) + a log(\|x + a - x) +--R (3) --------------------------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 10 +logmul1:=rule(c*log(a)+c*log(b) == c*log(a*b)) +--R +--I (4) c log(b) + c log(a) + %K == c log(a b) + %K +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 11 14:184 Schaums and Axiom differ by a constant +dd:=logmul1 cc +--R +--R 2 2 +--R a log(a ) +--R (5) --------- +--R 2 +--R Type: Expression Integer +--E + @ \section{\cite{1}:14.185~~~~~$\displaystyle @@ -83,7 +158,7 @@ $$ <<*>>= )clear all ---S 4 of 28 +--S 12 aa:=integrate(x^3/sqrt(x^2+a^2),x) --R --R @@ -96,6 +171,24 @@ aa:=integrate(x^3/sqrt(x^2+a^2),x) --R (12x + 3a )\|x + a - 12x - 9a x --R Type: Union(Expression Integer,...) --E + +--S 13 +bb:=(x^2+a^2)^(3/2)/3-a^2*sqrt(x^2+a^2) +--R +--R +-------+ +--R 2 2 | 2 2 +--R (x - 2a )\|x + a +--R (2) -------------------- +--R 3 +--R Type: Expression Integer +--E + +--S 14 14:185 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.186~~~~~$\displaystyle\int{\frac{dx}{x\sqrt{x^2+a^2}}}$} @@ -105,7 +198,7 @@ $$ <<*>>= )clear all ---S 5 of 28 +--S 15 aa:=integrate(1/(x*sqrt(x^2+a^2)),x) --R --R @@ -116,6 +209,57 @@ aa:=integrate(1/(x*sqrt(x^2+a^2)),x) --R a --R Type: Union(Expression Integer,...) --E + +--S 16 +bb:=-1/a*log((a+sqrt(x^2+a^2))/x) +--R +--R +-------+ +--R | 2 2 +--R \|x + a + a +--R log(--------------) +--R x +--R (2) - ------------------- +--R a +--R Type: Expression Integer +--E + +--S 17 +cc:=aa-bb +--R +--R (3) +--R +-------+ +--R +-------+ +-------+ | 2 2 +--R | 2 2 | 2 2 \|x + a + a +--R - log(\|x + a - x + a) + log(\|x + a - x - a) + log(--------------) +--R x +--R ------------------------------------------------------------------------- +--R a +--R Type: Expression Integer +--E + +--S 18 +divlog:=rule(log(a/b) == log(a) - log(b)) +--R +--R a +--R (4) log(-) == - log(b) + log(a) +--R b +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 19 14:186 Axiom cannot simplify this expression +dd:=divlog cc +--R +--R (5) +--R +-------+ +-------+ +-------+ +--R | 2 2 | 2 2 | 2 2 +--R log(\|x + a + a) - log(\|x + a - x + a) + log(\|x + a - x - a) +--R + +--R - log(x) +--R / +--R a +--R Type: Expression Integer +--E + @ \section{\cite{1}:14.187~~~~~$\displaystyle @@ -126,7 +270,7 @@ $$ <<*>>= )clear all ---S 6 of 28 +--S 20 aa:=integrate(1/(x^2*sqrt(x^2+a^2)),x) --R --R @@ -137,6 +281,29 @@ aa:=integrate(1/(x^2*sqrt(x^2+a^2)),x) --R x\|x + a - x --R Type: Union(Expression Integer,...) --E + +--S 21 +bb:=-sqrt(x^2+a^2)/(a^2*x) +--R +--R +-------+ +--R | 2 2 +--R \|x + a +--R (2) - ---------- +--R 2 +--R a x +--R Type: Expression Integer +--E + +--S 22 14:187 Schaums and Axiom differ by a constant +cc:=aa-bb +--R +--R 1 +--R (3) - -- +--R 2 +--R a +--R Type: Expression Integer +--E + @ \section{\cite{1}:14.188~~~~~$\displaystyle\int{\frac{dx}{x^3\sqrt{x^2+a^2}}}$} @@ -147,7 +314,7 @@ $$ <<*>>= )clear all ---S 7 of 28 +--S 23 aa:=integrate(1/(x^3*sqrt(x^2+a^2)),x) --R --R @@ -169,6 +336,59 @@ aa:=integrate(1/(x^3*sqrt(x^2+a^2)),x) --R 4a x \|x + a - 4a x - 2a x --R Type: Union(Expression Integer,...) --E + +--S 24 +bb:=-sqrt(x^2+a^2)/(2*a^2*x^2)+1/(2*a^3)*log((a+sqrt(x^2+a^2))/x) +--R +--R +-------+ +--R | 2 2 +-------+ +--R 2 \|x + a + a | 2 2 +--R x log(--------------) - a\|x + a +--R x +--R (2) ----------------------------------- +--R 3 2 +--R 2a x +--R Type: Expression Integer +--E + +--S 25 +cc:=aa-bb +--R +--R (3) +--R +-------+ +--R +-------+ +-------+ | 2 2 +--R | 2 2 | 2 2 \|x + a + a +--R log(\|x + a - x + a) - log(\|x + a - x - a) - log(--------------) +--R x +--R ----------------------------------------------------------------------- +--R 3 +--R 2a +--R Type: Expression Integer +--E + +--S 26 +divlog:=rule(log(a/b) == log(a) - log(b)) +--R +--R a +--R (4) log(-) == - log(b) + log(a) +--R b +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 27 14:188 Axiom cannot simplify this expression +dd:=divlog cc +--R +--R (5) +--R +-------+ +-------+ +-------+ +--R | 2 2 | 2 2 | 2 2 +--R - log(\|x + a + a) + log(\|x + a - x + a) - log(\|x + a - x - a) +--R + +--R log(x) +--R / +--R 3 +--R 2a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.189~~~~~$\displaystyle\int{\sqrt{x^2+a^2}}~dx$} @@ -178,7 +398,7 @@ $$ <<*>>= )clear all ---S 8 of 28 +--S 28 aa:=integrate(sqrt(x^2+a^2),x) --R --R @@ -196,6 +416,28 @@ aa:=integrate(sqrt(x^2+a^2),x) --R 4x\|x + a - 4x - 2a --R Type: Union(Expression Integer,...) --E + +--S 29 +bb:=(x*sqrt(x^2+a^2))/2+a^2/2*log(x+sqrt(x^2+a^2)) +--R +--R +-------+ +-------+ +--R 2 | 2 2 | 2 2 +--R a log(\|x + a + x) + x\|x + a +--R (2) ----------------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 30 14:189 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R +-------+ +-------+ +--R 2 | 2 2 2 | 2 2 +--R - a log(\|x + a + x) - a log(\|x + a - x) +--R (3) ----------------------------------------------- +--R 2 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.190~~~~~$\displaystyle\int{x\sqrt{x^2+a^2}}~dx$} @@ -205,7 +447,7 @@ $$ <<*>>= )clear all ---S 9 of 28 +--S 31 aa:=integrate(x*sqrt(x^2+a^2),x) --R --R @@ -218,6 +460,24 @@ aa:=integrate(x*sqrt(x^2+a^2),x) --R (12x + 3a )\|x + a - 12x - 9a x --R Type: Union(Expression Integer,...) --E + +--S 32 +bb:=(x^2+a^2)^(3/2)/3 +--R +--R +-------+ +--R 2 2 | 2 2 +--R (x + a )\|x + a +--R (2) ------------------- +--R 3 +--R Type: Expression Integer +--E + +--S 33 14:190 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.191~~~~~$\displaystyle @@ -229,7 +489,7 @@ $$ <<*>>= )clear all ---S 10 of 28 +--S 34 aa:=integrate(x^2*sqrt(x^2+a^2),x) --R --R @@ -247,6 +507,28 @@ aa:=integrate(x^2*sqrt(x^2+a^2),x) --R (64x + 32a x)\|x + a - 64x - 64a x - 8a --R Type: Union(Expression Integer,...) --E + +--S 35 +bb:=(x*(x^2+a^2)^(3/2))/4-(a^2*x*sqrt(x^2+a^2))/8-a^4/8*log(x+sqrt(x^2+a^2)) +--R +--R +-------+ +-------+ +--R 4 | 2 2 3 2 | 2 2 +--R - a log(\|x + a + x) + (2x + a x)\|x + a +--R (2) ----------------------------------------------- +--R 8 +--R Type: Expression Integer +--E + +--S 36 14:191 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R +-------+ +-------+ +--R 4 | 2 2 4 | 2 2 +--R a log(\|x + a + x) + a log(\|x + a - x) +--R (3) --------------------------------------------- +--R 8 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.192~~~~~$\displaystyle @@ -257,7 +539,7 @@ $$ <<*>>= )clear all ---S 11 of 28 +--S 37 aa:=integrate(x^3*sqrt(x^2+a^2),x) --R --R @@ -274,6 +556,24 @@ aa:=integrate(x^3*sqrt(x^2+a^2),x) --R (240x + 180a x + 15a )\|x + a - 240x - 300a x - 75a x --R Type: Union(Expression Integer,...) --E + +--S 38 +bb:=(x^2+a^2)^(5/2)/5-(a^2*(x^2+a^2)^(3/2))/3 +--R +--R +-------+ +--R 4 2 2 4 | 2 2 +--R (3x + a x - 2a )\|x + a +--R (2) ---------------------------- +--R 15 +--R Type: Expression Integer +--E + +--S 39 14:192 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.193~~~~~$\displaystyle @@ -284,7 +584,7 @@ $$ <<*>>= )clear all ---S 12 of 28 +--S 40 aa:=integrate(sqrt(x^2+a^2)/x,x) --R --R @@ -302,6 +602,33 @@ aa:=integrate(sqrt(x^2+a^2)/x,x) --R \|x + a - x --R Type: Union(Expression Integer,...) --E + +--S 41 +bb:=sqrt(x^2+a^2)-a*log((a+sqrt(x^2+a^2))/x) +--R +--R +-------+ +--R | 2 2 +-------+ +--R \|x + a + a | 2 2 +--R (2) - a log(--------------) + \|x + a +--R x +--R Type: Expression Integer +--E + +--S 42 14:193 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R (3) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R - a log(\|x + a - x + a) + a log(\|x + a - x - a) +--R + +--R +-------+ +--R | 2 2 +--R \|x + a + a +--R a log(--------------) +--R x +--R Type: Expression Integer +--E @ \section{\cite{1}:14.194~~~~~$\displaystyle @@ -312,7 +639,7 @@ $$ <<*>>= )clear all ---S 13 of 28 +--S 43 aa:=integrate(sqrt(x^2+a^2)/x^2,x) --R --R @@ -325,6 +652,26 @@ aa:=integrate(sqrt(x^2+a^2)/x^2,x) --R x\|x + a - x --R Type: Union(Expression Integer,...) --E + +--S 44 +bb:=-sqrt(x^2+a^2)/x+log(x+sqrt(x^2+a^2)) +--R +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R x log(\|x + a + x) - \|x + a +--R (2) ---------------------------------- +--R x +--R Type: Expression Integer +--E + +--S 45 14:194 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R (3) - log(\|x + a + x) - log(\|x + a - x) - 1 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.195~~~~~$\displaystyle @@ -336,7 +683,7 @@ $$ <<*>>= )clear all ---S 14 of 28 +--S 46 aa:=integrate(sqrt(x^2+a^2)/x^3,x) --R --R @@ -358,6 +705,57 @@ aa:=integrate(sqrt(x^2+a^2)/x^3,x) --R 4a x \|x + a - 4a x - 2a x --R Type: Union(Expression Integer,...) --E + +--S 47 +bb:=-sqrt(x^2+a^2)/(2*x^2)-1/(2*a)*log((a+sqrt(x^2+a^2))/x) +--R +--R +-------+ +--R | 2 2 +-------+ +--R 2 \|x + a + a | 2 2 +--R - x log(--------------) - a\|x + a +--R x +--R (2) ------------------------------------- +--R 2 +--R 2a x +--R Type: Expression Integer +--E + +--S 48 +cc:=aa-bb +--R +--R (3) +--R +-------+ +--R +-------+ +-------+ | 2 2 +--R | 2 2 | 2 2 \|x + a + a +--R - log(\|x + a - x + a) + log(\|x + a - x - a) + log(--------------) +--R x +--R ------------------------------------------------------------------------- +--R 2a +--R Type: Expression Integer +--E + +--S 49 +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 + +--S 50 14:195 Axiom cannot simplify this expression +dd:=divlog cc +--R +--R (5) +--R +-------+ +-------+ +-------+ +--R | 2 2 | 2 2 | 2 2 +--R log(\|x + a + a) - log(\|x + a - x + a) + log(\|x + a - x - a) +--R + +--R - log(x) +--R / +--R 2a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.196~~~~~$\displaystyle\int{\frac{dx}{(x^2+a^2)^{3/2}}}$} @@ -367,7 +765,7 @@ $$ <<*>>= )clear all ---S 15 of 28 +--S 51 aa:=integrate(1/(x^2+a^2)^(3/2),x) --R --R @@ -378,6 +776,27 @@ aa:=integrate(1/(x^2+a^2)^(3/2),x) --R x\|x + a - x - a --R Type: Union(Expression Integer,...) --E + +--S 52 +bb:=x/(a^2*sqrt(x^2+a^2)) +--R +--R x +--R (2) ------------ +--R +-------+ +--R 2 | 2 2 +--R a \|x + a +--R Type: Expression Integer +--E + +--S 53 14:196 Schaums and Axiom differ by a constant +cc:=aa-bb +--R +--R 1 +--R (3) -- +--R 2 +--R a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.197~~~~~$\displaystyle @@ -388,7 +807,7 @@ $$ <<*>>= )clear all ---S 16 of 28 +--S 54 aa:=integrate(x/(x^2+a^2)^(3/2),x) --R --R @@ -401,6 +820,25 @@ aa:=integrate(x/(x^2+a^2)^(3/2),x) --R x\|x + a - x - a --R Type: Union(Expression Integer,...) --E + +--S 55 +bb:=-1/sqrt(x^2+a^2) +--R +--R 1 +--R (2) - ---------- +--R +-------+ +--R | 2 2 +--R \|x + a +--R Type: Expression Integer +--E + +--S 56 14:197 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E + @ \section{\cite{1}:14.198~~~~~$\displaystyle @@ -411,7 +849,7 @@ $$ <<*>>= )clear all ---S 17 of 28 +--S 57 aa:=integrate(x^2/(x^2+a^2)^(3/2),x) --R --R @@ -424,6 +862,28 @@ aa:=integrate(x^2/(x^2+a^2)^(3/2),x) --R x\|x + a - x - a --R Type: Union(Expression Integer,...) --E + +--S 58 +bb:=-x/sqrt(x^2+a^2)+log(x+sqrt(x^2+a^2)) +--R +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R \|x + a log(\|x + a + x) - x +--R (2) --------------------------------- +--R +-------+ +--R | 2 2 +--R \|x + a +--R Type: Expression Integer +--E + +--S 59 14:198 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R (3) - log(\|x + a + x) - log(\|x + a - x) - 1 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.199~~~~~$\displaystyle @@ -434,7 +894,7 @@ $$ <<*>>= )clear all ---S 18 of 28 +--S 60 aa:=integrate(x^3/(x^2+a^2)^(3/2),x) --R --R @@ -447,6 +907,25 @@ aa:=integrate(x^3/(x^2+a^2)^(3/2),x) --R (2x + a )\|x + a - 2x - 2a x --R Type: Union(Expression Integer,...) --E + +--S 61 +bb:=sqrt(x^2+a^2)+a^2/sqrt(x^2+a^2) +--R +--R 2 2 +--R x + 2a +--R (2) ---------- +--R +-------+ +--R | 2 2 +--R \|x + a +--R Type: Expression Integer +--E + +--S 62 14:199 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.200~~~~~$\displaystyle @@ -458,7 +937,7 @@ $$ <<*>>= )clear all ---S 19 of 28 +--S 63 aa:=integrate(1/(x*(x^2+a^2)^(3/2)),x) --R --R @@ -476,6 +955,60 @@ aa:=integrate(1/(x*(x^2+a^2)^(3/2)),x) --R a x\|x + a - a x - a --R Type: Union(Expression Integer,...) --E + +--S 64 +bb:=1/(a^2*sqrt(x^2+a^2))-1/a^3*log((a+sqrt(x^2+a^2))/x) +--R +--R +-------+ +--R +-------+ | 2 2 +--R | 2 2 \|x + a + a +--R - \|x + a log(--------------) + a +--R x +--R (2) ----------------------------------- +--R +-------+ +--R 3 | 2 2 +--R a \|x + a +--R Type: Expression Integer +--E + +--S 65 +cc:=aa-bb +--R +--R (3) +--R +-------+ +--R +-------+ +-------+ | 2 2 +--R | 2 2 | 2 2 \|x + a + a +--R - log(\|x + a - x + a) + log(\|x + a - x - a) + log(--------------) +--R x +--R ------------------------------------------------------------------------- +--R 3 +--R a +--R Type: Expression Integer +--E + +--S 66 +divlog:=rule(log(a/b) == log(a) - log(b)) +--R +--R a +--R (4) log(-) == - log(b) + log(a) +--R b +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 67 14:200 Axiom cannot simplify this expression +dd:=divlog cc +--R +--R (5) +--R +-------+ +-------+ +-------+ +--R | 2 2 | 2 2 | 2 2 +--R log(\|x + a + a) - log(\|x + a - x + a) + log(\|x + a - x - a) +--R + +--R - log(x) +--R / +--R 3 +--R a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.201~~~~~$\displaystyle @@ -486,7 +1019,7 @@ $$ <<*>>= )clear all ---S 20 of 28 +--S 68 aa:=integrate(1/(x^2*(x^2+a^2)^(3/2)),x) --R --R @@ -497,6 +1030,28 @@ aa:=integrate(1/(x^2*(x^2+a^2)^(3/2)),x) --R (2x + a x)\|x + a - 2x - 2a x --R Type: Union(Expression Integer,...) --E + +--S 69 +bb:=-sqrt(x^2+a^2)/(a^4*x)-x/(a^4*sqrt(x^2+a^2)) +--R +--R 2 2 +--R - 2x - a +--R (2) ------------- +--R +-------+ +--R 4 | 2 2 +--R a x\|x + a +--R Type: Expression Integer +--E + +--S 70 14:201 Schaums and Axiom differ by a constant +cc:=aa-bb +--R +--R 2 +--R (3) - -- +--R 4 +--R a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.202~~~~~$\displaystyle @@ -509,7 +1064,7 @@ $$ <<*>>= )clear all ---S 21 of 28 +--S 71 aa:=integrate(1/(x^3*(x^2+a^2)^(3/2)),x) --R --R @@ -535,6 +1090,62 @@ aa:=integrate(1/(x^3*(x^2+a^2)^(3/2)),x) --R (8a x + 6a x )\|x + a - 8a x - 10a x - 2a x --R Type: Union(Expression Integer,...) --E + +--S 72 +bb:=-1/(2*a^2*x^2*sqrt(x^2+a^2))-3/(2*a^4*sqrt(x^2+a^2))+3/(2*a^5)*log((a+sqrt(x^2+a^2))/x) +--R +--R +-------+ +--R +-------+ | 2 2 +--R 2 | 2 2 \|x + a + a 2 3 +--R 3x \|x + a log(--------------) - 3a x - a +--R x +--R (2) --------------------------------------------- +--R +-------+ +--R 5 2 | 2 2 +--R 2a x \|x + a +--R Type: Expression Integer +--E + +--S 73 +cc:=aa-bb +--R +--R (3) +--R +-------+ +--R +-------+ +-------+ | 2 2 +--R | 2 2 | 2 2 \|x + a + a +--R 3log(\|x + a - x + a) - 3log(\|x + a - x - a) - 3log(--------------) +--R x +--R -------------------------------------------------------------------------- +--R 5 +--R 2a +--R Type: Expression Integer +--E + +--S 74 +divlog:=rule(log(a/b) == log(a) - log(b)) +--R +--R a +--R (4) log(-) == - log(b) + log(a) +--R b +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 75 14:202 Axiom cannot simplify this expression +dd:=divlog cc +--R +--R (5) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R - 3log(\|x + a + a) + 3log(\|x + a - x + a) +--R + +--R +-------+ +--R | 2 2 +--R - 3log(\|x + a - x - a) + 3log(x) +--R / +--R 5 +--R 2a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.203~~~~~$\displaystyle\int{(x^2+a^2)^{3/2}}~dx$} @@ -545,9 +1156,8 @@ $$ <<*>>= )clear all ---S 22 of 28 +--S 76 aa:=integrate((x^2+a^2)^(3/2),x) ---R --R --R (1) --R +-------+ +-------+ @@ -566,6 +1176,28 @@ aa:=integrate((x^2+a^2)^(3/2),x) --R (64x + 32a x)\|x + a - 64x - 64a x - 8a --R Type: Union(Expression Integer,...) --E + +--S 77 +bb:=(x*(x^2+a^2)^(3/2))/4+(3*a^2*x*sqrt(x^2+a^2))/8+3/8*a^4*log(x+sqrt(x^2+a^2)) +--R +--R +-------+ +-------+ +--R 4 | 2 2 3 2 | 2 2 +--R 3a log(\|x + a + x) + (2x + 5a x)\|x + a +--R (2) ----------------------------------------------- +--R 8 +--R Type: Expression Integer +--E + +--S 78 14:203 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R +-------+ +-------+ +--R 4 | 2 2 4 | 2 2 +--R - 3a log(\|x + a + x) - 3a log(\|x + a - x) +--R (3) ------------------------------------------------- +--R 8 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.204~~~~~$\displaystyle\int{x(x^2+a^2)^{3/2}}~dx$} @@ -573,7 +1205,7 @@ $$\int{x(x^2+a^2)^{3/2}}=\frac{(x^2+a^2)^{5/2}}{5}$$ <<*>>= )clear all ---S 23 of 28 +--S 79 aa:=integrate(x*(x^2+a^2)^(3/2),x) --R --R @@ -590,6 +1222,24 @@ aa:=integrate(x*(x^2+a^2)^(3/2),x) --R (80x + 60a x + 5a )\|x + a - 80x - 100a x - 25a x --R Type: Union(Expression Integer,...) --E + +--S 80 +bb:=(x^2+a^2)^(5/2)/5 +--R +--R +-------+ +--R 4 2 2 4 | 2 2 +--R (x + 2a x + a )\|x + a +--R (2) --------------------------- +--R 5 +--R Type: Expression Integer +--E + +--S 81 14:204 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.205~~~~~$\displaystyle\int{x^2(x^2+a^2)^{3/2}}~dx$} @@ -601,7 +1251,7 @@ $$ <<*>>= )clear all ---S 24 of 28 +--S 82 aa:=integrate(x^2*(x^2+a^2)^(3/2),x) --R --R @@ -629,6 +1279,28 @@ aa:=integrate(x^2*(x^2+a^2)^(3/2),x) --R (1536x + 1536a x + 288a x)\|x + a - 1536x - 2304a x - 864a x - 48a --R Type: Union(Expression Integer,...) --E + +--S 83 +bb:=(x*(x^2+a^2)^(5/2))/6-(a^2*x*(x^2+a^2)^(3/2))/24-(a^4*x*sqrt(x^2+a^2))/16-a^6/16*log(x+sqrt(x^2+a^2)) +--R +--R +-------+ +-------+ +--R 6 | 2 2 5 2 3 4 | 2 2 +--R - 3a log(\|x + a + x) + (8x + 14a x + 3a x)\|x + a +--R (2) ---------------------------------------------------------- +--R 48 +--R Type: Expression Integer +--E + +--S 84 14:205 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R +-------+ +-------+ +--R 6 | 2 2 6 | 2 2 +--R a log(\|x + a + x) + a log(\|x + a - x) +--R (3) --------------------------------------------- +--R 16 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.206~~~~~$\displaystyle\int{x^3(x^2+a^2)^{3/2}}~dx$} @@ -638,7 +1310,7 @@ $$ <<*>>= )clear all ---S 25 of 28 +--S 85 aa:=integrate(x^3*(x^2+a^2)^(3/2),x) --R --R @@ -667,6 +1339,24 @@ aa:=integrate(x^3*(x^2+a^2)^(3/2),x) --R - 1960a x - 245a x --R Type: Union(Expression Integer,...) --E + +--S 86 +bb:=(x^2+a^2)^(7/2)/7-(a^2*(x^2+a^2)^(5/2))/5 +--R +--R +-------+ +--R 6 2 4 4 2 6 | 2 2 +--R (5x + 8a x + a x - 2a )\|x + a +--R (2) ------------------------------------ +--R 35 +--R Type: Expression Integer +--E + +--S 87 14:206 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.207~~~~~$\displaystyle @@ -678,7 +1368,7 @@ $$ <<*>>= )clear all ---S 26 of 28 +--S 88 aa:=integrate((x^2+a^2)^(3/2)/x,x) --R --R @@ -700,6 +1390,58 @@ aa:=integrate((x^2+a^2)^(3/2)/x,x) --R (12x + 3a )\|x + a - 12x - 9a x --R Type: Union(Expression Integer,...) --E + +--S 89 +bb:=(x^2+a^2)^(3/2)/3+a^2*sqrt(x^2+a^2)-a^3*log((a+sqrt(x^2+a^2))/x) +--R +--R +-------+ +--R | 2 2 +-------+ +--R 3 \|x + a + a 2 2 | 2 2 +--R - 3a log(--------------) + (x + 4a )\|x + a +--R x +--R (2) ----------------------------------------------- +--R 3 +--R Type: Expression Integer +--E + +--S 90 +cc:=aa-bb +--R +--R (3) +--R +-------+ +-------+ +--R 3 | 2 2 3 | 2 2 +--R - a log(\|x + a - x + a) + a log(\|x + a - x - a) +--R + +--R +-------+ +--R | 2 2 +--R 3 \|x + a + a +--R a log(--------------) +--R x +--R Type: Expression Integer +--E + +--S 91 +divlog:=rule(log(a/b) == log(a) - log(b)) +--R +--R a +--R (4) log(-) == - log(b) + log(a) +--R b +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 92 14:207 Axiom cannot simplify this expression +dd:=divlog cc +--R +--R (5) +--R +-------+ +-------+ +--R 3 | 2 2 3 | 2 2 +--R a log(\|x + a + a) - a log(\|x + a - x + a) +--R + +--R +-------+ +--R 3 | 2 2 3 +--R a log(\|x + a - x - a) - a log(x) +--R Type: Expression Integer +--E @ \section{\cite{1}:14.208~~~~~$\displaystyle @@ -711,7 +1453,7 @@ $$ <<*>>= )clear all ---S 27 of 28 +--S 93 aa:=integrate((x^2+a^2)^{3/2}/x^2,x) --R --R @@ -729,6 +1471,28 @@ aa:=integrate((x^2+a^2)^{3/2}/x^2,x) --R (8x + 2a x)\|x + a - 8x - 6a x --R Type: Union(Expression Integer,...) --E + +--S 94 +bb:=-(x^2+a^2)^(3/2)/x+(3*x*sqrt(x^2+a^2))/2+3/2*a^2*log(x+sqrt(x^2+a^2)) +--R +--R +-------+ +-------+ +--R 2 | 2 2 2 2 | 2 2 +--R 3a x log(\|x + a + x) + (x - 2a )\|x + a +--R (2) ----------------------------------------------- +--R 2x +--R Type: Expression Integer +--E + +--S 95 14:208 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R +-------+ +-------+ +--R 2 | 2 2 2 | 2 2 2 +--R - 3a log(\|x + a + x) - 3a log(\|x + a - x) - 2a +--R (3) ------------------------------------------------------- +--R 2 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.209~~~~~$\displaystyle @@ -740,7 +1504,7 @@ $$ <<*>>= )clear all ---S 28 of 28 +--S 96 aa:=integrate((x^2+a^2)^(3/2)/x^3,x) --R --R @@ -763,6 +1527,51 @@ aa:=integrate((x^2+a^2)^(3/2)/x^3,x) --R Type: Union(Expression Integer,...) --E +--S 97 +bb:=-(x^2+a^2)^(3/2)/(2*x^3)+3/2*sqrt(x^2+a^2)-3/2*a*log((a+sqrt(x^2+a^2))/x) +--R +--R +-------+ +--R | 2 2 +-------+ +--R 3 \|x + a + a 3 2 2 | 2 2 +--R - 3a x log(--------------) + (3x - x - a )\|x + a +--R x +--R (2) ------------------------------------------------------ +--R 3 +--R 2x +--R Type: Expression Integer +--E + +--S 98 14:209 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R (3) +--R +-------+ +-------+ +--R 5 3 3 | 2 2 6 3 4 | 2 2 +--R ((- 12a x - 3a x )\|x + a + 12a x + 9a x )log(\|x + a - x + a) +--R + +--R +-------+ +-------+ +--R 5 3 3 | 2 2 6 3 4 | 2 2 +--R ((12a x + 3a x )\|x + a - 12a x - 9a x )log(\|x + a - x - a) +--R + +--R +-------+ +--R +-------+ | 2 2 +--R 5 3 3 | 2 2 6 3 4 \|x + a + a +--R ((12a x + 3a x )\|x + a - 12a x - 9a x )log(--------------) +--R x +--R + +--R +-------+ +--R 6 5 2 4 2 3 4 2 4 | 2 2 7 6 2 5 +--R (4x - 4x + 7a x - 7a x + 3a x - 3a x)\|x + a - 4x + 4x - 9a x +--R + +--R 2 4 4 3 4 2 6 6 +--R 9a x - 6a x + 6a x - a x + a +--R / +--R +-------+ +--R 5 2 3 | 2 2 6 2 4 +--R (8x + 2a x )\|x + a - 8x - 6a x +--R Type: Expression Integer +--E + )spool )lisp (bye) @