diff --git a/changelog b/changelog index 314b281..3956a7b 100644 --- a/changelog +++ b/changelog @@ -1,3 +1,5 @@ +20080425 tpd src/input/schaum25.input show Schaums-Axiom equivalence +20080425 tpd src/input/schaum24.input show Schaums-Axiom equivalence 20080424 tpd src/input/schaum23.input show Schaums-Axiom equivalence 20080423 tpd src/input/schaum22.input show Schaums-Axiom equivalence 20080423 tpd src/input/schaum21.input show Schaums-Axiom equivalence diff --git a/src/input/schaum24.input.pamphlet b/src/input/schaum24.input.pamphlet index 1a6b60a..9c034da 100644 --- a/src/input/schaum24.input.pamphlet +++ b/src/input/schaum24.input.pamphlet @@ -18,7 +18,7 @@ $$ )set message auto off )clear all ---S 1 of 38 +--S 1 aa:=integrate(asin(x/a),x) --R --R @@ -32,6 +32,31 @@ aa:=integrate(asin(x/a),x) --R 2 --R Type: Union(Expression Integer,...) --E + +--S 2 +bb:=s+asin(x/a)+sqrt(a^2-x^2) +--R +--R +---------+ +--R | 2 2 x +--R (2) \|- x + a + asin(-) + s +--R a +--R Type: Expression Integer +--E + +--S 3 14:471 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R +---------+ +--R | 2 2 +--R 2x\|- x + a x +--R - x atan(--------------) - 2asin(-) - 2s +--R 2 2 a +--R 2x - a +--R (3) ---------------------------------------- +--R 2 +--R Type: Expression Integer +--E + @ \section{\cite{1}:14.472~~~~~$\displaystyle @@ -43,7 +68,7 @@ $$ <<*>>= )clear all ---S 2 of 38 +--S 4 aa:=integrate(x*asin(x/a),x) --R --R @@ -57,6 +82,32 @@ aa:=integrate(x*asin(x/a),x) --R 8 --R Type: Union(Expression Integer,...) --E + +--S 5 +bb:=(x^2/2-a^2/4)*asin(x/a)+(x*sqrt(a^2-x^2))/4 +--R +--R +---------+ +--R | 2 2 2 2 x +--R x\|- x + a + (2x - a )asin(-) +--R a +--R (2) --------------------------------- +--R 4 +--R Type: Expression Integer +--E + +--S 6 14:472 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R +---------+ +--R | 2 2 +--R 2 2 2x\|- x + a 2 2 x +--R (- 2x + a )atan(--------------) + (- 4x + 2a )asin(-) +--R 2 2 a +--R 2x - a +--R (3) ------------------------------------------------------- +--R 8 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.473~~~~~$\displaystyle @@ -67,7 +118,7 @@ $$ <<*>>= )clear all ---S 3 of 38 +--S 7 aa:=integrate(x^2*asin(x/a),x) --R --R @@ -81,6 +132,32 @@ aa:=integrate(x^2*asin(x/a),x) --R 18 --R Type: Union(Expression Integer,...) --E + +--S 8 +bb:=x^3/3*asin(x/a)+((x^2+2*a^2)*sqrt(a^2-x^2))/9 +--R +--R +---------+ +--R 2 2 | 2 2 3 x +--R (x + 2a )\|- x + a + 3x asin(-) +--R a +--R (2) ----------------------------------- +--R 9 +--R Type: Expression Integer +--E + +--S 9 14:473 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R +---------+ +--R | 2 2 +--R 3 2x\|- x + a 3 x +--R - x atan(--------------) - 2x asin(-) +--R 2 2 a +--R 2x - a +--R (3) ------------------------------------- +--R 6 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.474~~~~~$\displaystyle @@ -93,7 +170,7 @@ $$ <<*>>= )clear all ---S 4 of 38 +--S 10 14:474 Axiom cannot compute this integral aa:=integrate(asin(x/a)/x,x) --R --R @@ -115,7 +192,7 @@ $$ <<*>>= )clear all ---S 5 of 38 +--S 11 aa:=integrate(asin(x/a)/x^2,x) --R --R @@ -130,6 +207,38 @@ aa:=integrate(asin(x/a)/x^2,x) --R 2a x --R Type: Union(Expression Integer,...) --E + +--S 12 +bb:=-asin(x/a)/x-1/a*log((a+sqrt(a^2-x^2))/x) +--R +--R +---------+ +--R | 2 2 +--R \|- x + a + a x +--R - x log(----------------) - a asin(-) +--R x a +--R (2) ------------------------------------- +--R a x +--R Type: Expression Integer +--E + +--S 13 14:475 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R (3) +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R - x log(\|- x + a + a) + x log(\|- x + a - a) +--R + +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R \|- x + a + a 2x\|- x + a x +--R 2x log(----------------) + a atan(--------------) + 2a asin(-) +--R x 2 2 a +--R 2x - a +--R / +--R 2a x +--R Type: Expression Integer +--E @ \section{\cite{1}:14.476~~~~~$\displaystyle @@ -140,7 +249,7 @@ $$ <<*>>= )clear all ---S 6 of 38 +--S 14 aa:=integrate(asin(x/a)^2,x) --R --R @@ -154,6 +263,36 @@ aa:=integrate(asin(x/a)^2,x) --R 4 --R Type: Union(Expression Integer,...) --E + +--S 15 +bb:=x*asin(x/a)^2-2*x+2*sqrt(a^2-x^2)*asin(x/a) +--R +--R +---------+ +--R x | 2 2 x 2 +--R (2) 2asin(-)\|- x + a + x asin(-) - 2x +--R a a +--R Type: Expression Integer +--E + +--S 16 14:476 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R (3) +--R +---------+ 2 +---------+ +--R | 2 2 +---------+ | 2 2 +--R 2x\|- x + a | 2 2 2x\|- x + a +--R x atan(--------------) - 4\|- x + a atan(--------------) +--R 2 2 2 2 +--R 2x - a 2x - a +--R + +--R +---------+ +--R x | 2 2 x 2 +--R - 8asin(-)\|- x + a - 4x asin(-) +--R a a +--R / +--R 4 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.477~~~~~$\displaystyle @@ -164,7 +303,7 @@ $$ <<*>>= )clear all ---S 7 of 38 +--S 17 aa:=integrate(acos(x/a),x) --R --R @@ -178,6 +317,30 @@ aa:=integrate(acos(x/a),x) --R 2 --R Type: Union(Expression Integer,...) --E + +--S 18 +bb:=x*acos(x/a)-sqrt(a^2-x^2) +--R +--R +---------+ +--R | 2 2 x +--R (2) - \|- x + a + x acos(-) +--R a +--R Type: Expression Integer +--E + +--S 19 14:477 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R +---------+ +--R | 2 2 +--R 2x\|- x + a x +--R x atan(--------------) - 2x acos(-) +--R 2 2 a +--R 2x - a +--R (3) ----------------------------------- +--R 2 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.478~~~~~$\displaystyle @@ -189,7 +352,7 @@ $$ <<*>>= )clear all ---S 8 of 38 +--S 20 aa:=integrate(x*acos(x/a),x) --R --R @@ -203,6 +366,32 @@ aa:=integrate(x*acos(x/a),x) --R 8 --R Type: Union(Expression Integer,...) --E + +--S 21 +bb:=(x^2/2-a^2/4)*acos(x/a)-(x*sqrt(a^2-x^2))/4 +--R +--R +---------+ +--R | 2 2 2 2 x +--R - x\|- x + a + (2x - a )acos(-) +--R a +--R (2) ----------------------------------- +--R 4 +--R Type: Expression Integer +--E + +--S 22 14:478 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R +---------+ +--R | 2 2 +--R 2 2 2x\|- x + a 2 2 x +--R (2x - a )atan(--------------) + (- 4x + 2a )acos(-) +--R 2 2 a +--R 2x - a +--R (3) ----------------------------------------------------- +--R 8 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.479~~~~~$\displaystyle @@ -213,7 +402,7 @@ $$ <<*>>= )clear all ---S 9 of 38 +--S 23 aa:=integrate(x^2*acos(x/a),x) --R --R @@ -227,6 +416,32 @@ aa:=integrate(x^2*acos(x/a),x) --R 18 --R Type: Union(Expression Integer,...) --E + +--S 24 +bb:=x^3/3*acos(x/a)-((x^2+2*a^2)*sqrt(a^2-x^2))/9 +--R +--R +---------+ +--R 2 2 | 2 2 3 x +--R (- x - 2a )\|- x + a + 3x acos(-) +--R a +--R (2) ------------------------------------- +--R 9 +--R Type: Expression Integer +--E + +--S 25 14:479 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R +---------+ +--R | 2 2 +--R 3 2x\|- x + a 3 x +--R x atan(--------------) - 2x acos(-) +--R 2 2 a +--R 2x - a +--R (3) ----------------------------------- +--R 6 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.480~~~~~$\displaystyle @@ -237,7 +452,7 @@ $$ <<*>>= )clear all ---S 10 of 38 +--S 26 14:480 Axiom cannot compute this integral aa:=integrate(acos(x/a)/x,x) --R --R @@ -258,7 +473,7 @@ $$ <<*>>= )clear all ---S 11 of 38 +--S 27 aa:=integrate(acos(x/a)/x^2,x) --R --R @@ -273,6 +488,38 @@ aa:=integrate(acos(x/a)/x^2,x) --R 2a x --R Type: Union(Expression Integer,...) --E + +--S 28 +bb:=-acos(x/a)/x+1/a*log((a+sqrt(a^2-x^2))/x) +--R +--R +---------+ +--R | 2 2 +--R \|- x + a + a x +--R x log(----------------) - a acos(-) +--R x a +--R (2) ----------------------------------- +--R a x +--R Type: Expression Integer +--E + +--S 29 14:481 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R (3) +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R x log(\|- x + a + a) - x log(\|- x + a - a) +--R + +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R \|- x + a + a 2x\|- x + a x +--R - 2x log(----------------) - a atan(--------------) + 2a acos(-) +--R x 2 2 a +--R 2x - a +--R / +--R 2a x +--R Type: Expression Integer +--E @ \section{\cite{1}:14.482~~~~~$\displaystyle @@ -283,7 +530,7 @@ $$ <<*>>= )clear all ---S 12 of 38 +--S 30 aa:=integrate(acos(x/a)^2,x) --R --R @@ -296,7 +543,37 @@ aa:=integrate(acos(x/a)^2,x) --R (1) ---------------------------------------------------------------- --R 4 --R Type: Union(Expression Integer,...) ---E +--E + +--S 31 +bb:=x*acos(x/a)^2-2*x-2*sqrt(a^2-x^2)*acos(x/a) +--R +--R +---------+ +--R x | 2 2 x 2 +--R (2) - 2acos(-)\|- x + a + x acos(-) - 2x +--R a a +--R Type: Expression Integer +--E + +--S 32 14:482 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R (3) +--R +---------+ 2 +---------+ +--R | 2 2 +---------+ | 2 2 +--R 2x\|- x + a | 2 2 2x\|- x + a +--R x atan(--------------) - 4\|- x + a atan(--------------) +--R 2 2 2 2 +--R 2x - a 2x - a +--R + +--R +---------+ +--R x | 2 2 x 2 +--R 8acos(-)\|- x + a - 4x acos(-) +--R a a +--R / +--R 4 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.483~~~~~$\displaystyle @@ -307,7 +584,7 @@ $$ <<*>>= )clear all ---S 13 of 38 +--S 33 aa:=integrate(atan(x/a),x) --R --R @@ -319,6 +596,62 @@ aa:=integrate(atan(x/a),x) --R 2 --R Type: Union(Expression Integer,...) --E + +--S 34 +bb:=x*atan(x/a)-a/2*log(x^2+a^2) +--R +--R 2 2 x +--R - a log(x + a ) + 2x atan(-) +--R a +--R (2) ----------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 35 +cc:=aa-bb +--R +--R x 2a x +--R - 2x atan(-) - x atan(-------) +--R a 2 2 +--R x - a +--R (3) ------------------------------ +--R 2 +--R Type: Expression Integer +--E + +--S 36 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (4) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 37 +dd:=atanrule cc +--R +--R 2 2 +--R x + 2%i a x - a - x + %i a +--R %i x log(-----------------) + 2%i x log(----------) +--R 2 2 x + %i a +--R x - 2%i a x - a +--R (5) --------------------------------------------------- +--R 4 +--R Type: Expression Complex Integer +--E + +--S 38 14:483 SCHAUMS AND AXIOM DIFFER? (BRANCH CUTS?) +ee:=expandLog dd +--R +--R %i x log(- 1) +--R (6) ------------- +--R 2 +--R Type: Expression Complex Integer +--E @ \section{\cite{1}:14.484~~~~~$\displaystyle @@ -329,7 +662,7 @@ $$ <<*>>= )clear all ---S 14 of 38 +--S 39 14:484 Axiom cannot compute this integral aa:=integrate(x*tan(x/a),x) --R --R @@ -349,7 +682,7 @@ $$ <<*>>= )clear all ---S 15 of 38 +--S 40 aa:=integrate(x^2*atan(x/a),x) --R --R @@ -361,6 +694,29 @@ aa:=integrate(x^2*atan(x/a),x) --R 6 --R Type: Union(Expression Integer,...) --E + +--S 41 +bb:=x^3/2*atan(x/a)-(a*x^2)/6+a^3/6*log(x^2+a^2) +--R +--R 3 2 2 3 x 2 +--R a log(x + a ) + 3x atan(-) - a x +--R a +--R (2) ---------------------------------- +--R 6 +--R Type: Expression Integer +--E + +--S 42 14:485 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R 3 x 3 2a x +--R - 3x atan(-) - x atan(-------) +--R a 2 2 +--R x - a +--R (3) ------------------------------ +--R 6 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.486~~~~~$\displaystyle @@ -371,7 +727,7 @@ $$ <<*>>= )clear all ---S 16 of 38 +--S 43 14:486 Axiom cannot compute this integral aa:=integrate(atan(x/a)/x,x) --R --R @@ -393,7 +749,7 @@ $$ <<*>>= )clear all ---S 17 of 38 +--S 44 aa:=integrate(atan(x/a)/x^2,x) --R --R @@ -405,6 +761,73 @@ aa:=integrate(atan(x/a)/x^2,x) --R 2a x --R Type: Union(Expression Integer,...) --E + +--S 45 +bb:=-1/x*atan(x/a)-1/(2*a)*log((x^2+a^2)/x^2) +--R +--R 2 2 +--R x + a x +--R - x log(-------) - 2a atan(-) +--R 2 a +--R x +--R (2) ----------------------------- +--R 2a x +--R Type: Expression Integer +--E + +--S 46 +cc:=aa-bb +--R +--R (3) +--R 2 2 +--R 2 2 x + a x 2a x +--R - x log(x + a ) + 2x log(x) + x log(-------) + 2a atan(-) + a atan(-------) +--R 2 a 2 2 +--R x x - a +--R ---------------------------------------------------------------------------- +--R 2a x +--R Type: Expression Integer +--E + +--S 47 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (4) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 48 +dd:=atanrule cc +--R +--R (5) +--R 2 2 +--R 2 2 x + 2%i a x - a +--R - 2x log(x + a ) + 4x log(x) - %i a log(-----------------) +--R 2 2 +--R x - 2%i a x - a +--R + +--R 2 2 +--R x + a - x + %i a +--R 2x log(-------) - 2%i a log(----------) +--R 2 x + %i a +--R x +--R / +--R 4a x +--R Type: Expression Complex Integer +--E + +--S 49 14:487 SCHAUMS AND AXIOM DIFFER? (branch cuts?) +ee:=expandLog dd +--R +--R %i log(- 1) +--R (6) - ----------- +--R 2x +--R Type: Expression Complex Integer +--E @ \section{\cite{1}:14.488~~~~~$\displaystyle @@ -415,7 +838,7 @@ $$ <<*>>= )clear all ---S 18 of 38 +--S 50 aa:=integrate(acot(x/a),x) --R --R @@ -426,7 +849,85 @@ aa:=integrate(acot(x/a),x) --R (1) -------------------------------- --R 2 --R Type: Union(Expression Integer,...) +--E + +--S 51 +bb:=x*acot(x/a)+a/2*log(x^2+a^2) +--R +--R 2 2 x +--R a log(x + a ) + 2x acot(-) +--R a +--R (2) --------------------------- +--R 2 +--R Type: Expression Integer --E + +--S 52 +cc:=aa-bb +--R +--R 2a x x +--R x atan(-------) - 2x acot(-) +--R 2 2 a +--R x - a +--R (3) ---------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 53 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (4) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 54 +dd:=atanrule cc +--R +--R 2 2 +--R x + 2%i a x - a x +--R - %i x log(-----------------) - 4x acot(-) +--R 2 2 a +--R x - 2%i a x - a +--R (5) ------------------------------------------ +--R 4 +--R Type: Expression Complex Integer +--E + +--S 55 +acotrule:=rule(acot(x) == -%i/2*log((%i*x-1)/(%i*x+1))) +--R +--R x + %i +--R %i log(------) +--R x - %i +--R (6) acot(x) == - -------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 56 +ee:=acotrule dd +--R +--R 2 2 +--R x + 2%i a x - a x + %i a +--R - %i x log(-----------------) + 2%i x log(--------) +--R 2 2 x - %i a +--R x - 2%i a x - a +--R (7) --------------------------------------------------- +--R 4 +--R Type: Expression Complex Integer +--E + +--S 57 14:488 Axiom and Schaums agree +ff:=expandLog % +--R +--R (8) 0 +--R Type: Expression Complex Integer +--E @ \section{\cite{1}:14.489~~~~~$\displaystyle @@ -437,7 +938,7 @@ $$ <<*>>= )clear all ---S 19 of 38 +--S 58 aa:=integrate(x*acot(x/a),x) --R --R @@ -449,6 +950,84 @@ aa:=integrate(x*acot(x/a),x) --R 4 --R Type: Union(Expression Integer,...) --E + +--S 59 +bb:=1/2*(x^2+a^2)*acot(x/a)+(a*x)/2 +--R +--R 2 2 x +--R (x + a )acot(-) + a x +--R a +--R (2) ---------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 60 +cc:=aa-bb +--R +--R 2 2 2a x 2 2 x +--R (x + a )atan(-------) + (- 2x - 2a )acot(-) +--R 2 2 a +--R x - a +--R (3) --------------------------------------------- +--R 4 +--R Type: Expression Integer +--E + +--S 61 +acotrule:=rule(acot(x) == -%i/2*log((%i*x-1)/(%i*x+1))) +--R +--R x + %i +--R %i log(------) +--R x - %i +--R (4) acot(x) == - -------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 62 +dd:=acotrule cc +--R +--R 2 2 x + %i a 2 2 2a x +--R (%i x + %i a )log(--------) + (x + a )atan(-------) +--R x - %i a 2 2 +--R x - a +--R (5) ----------------------------------------------------- +--R 4 +--R Type: Expression Complex Integer +--E + +--S 63 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (6) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 64 +ee:=atanrule dd +--R +--R (7) +--R 2 2 +--R 2 2 x + 2%i a x - a 2 2 x + %i a +--R (- %i x - %i a )log(-----------------) + (2%i x + 2%i a )log(--------) +--R 2 2 x - %i a +--R x - 2%i a x - a +--R ------------------------------------------------------------------------ +--R 8 +--R Type: Expression Complex Integer +--E + +--S 65 14:489 Axiom and Schaums agree +ff:=expandLog ee +--R +--R (8) 0 +--R Type: Expression Complex Integer +--E @ \section{\cite{1}:14.490~~~~~$\displaystyle @@ -459,7 +1038,7 @@ $$ <<*>>= )clear all ---S 20 of 38 +--S 66 aa:=integrate(x^2*acot(x/a),x) --R --R @@ -470,7 +1049,85 @@ aa:=integrate(x^2*acot(x/a),x) --R (1) ----------------------------------------- --R 6 --R Type: Union(Expression Integer,...) +--E + +--S 67 +bb:=x^3/3*acot(x/a)+(a*x^2)/6-a^3/6*log(x^2+a^2) +--R +--R 3 2 2 3 x 2 +--R - a log(x + a ) + 2x acot(-) + a x +--R a +--R (2) ------------------------------------ +--R 6 +--R Type: Expression Integer --E + +--S 68 +cc:=aa-bb +--R +--R 3 2a x 3 x +--R x atan(-------) - 2x acot(-) +--R 2 2 a +--R x - a +--R (3) ---------------------------- +--R 6 +--R Type: Expression Integer +--E + +--S 69 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (4) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 70 +dd:=atanrule cc +--R +--R 2 2 +--R 3 x + 2%i a x - a 3 x +--R - %i x log(-----------------) - 4x acot(-) +--R 2 2 a +--R x - 2%i a x - a +--R (5) ------------------------------------------ +--R 12 +--R Type: Expression Complex Integer +--E + +--S 71 +acotrule:=rule(acot(x) == -%i/2*log((%i*x-1)/(%i*x+1))) +--R +--R x + %i +--R %i log(------) +--R x - %i +--R (6) acot(x) == - -------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 72 +ee:=acotrule dd +--R +--R 2 2 +--R 3 x + 2%i a x - a 3 x + %i a +--R - %i x log(-----------------) + 2%i x log(--------) +--R 2 2 x - %i a +--R x - 2%i a x - a +--R (7) --------------------------------------------------- +--R 12 +--R Type: Expression Complex Integer +--E + +--S 73 14:490 Axiom and Schaums agree +ff:=expandLog ee +--R +--R (8) 0 +--R Type: Expression Complex Integer +--E @ \section{\cite{1}:14.491~~~~~$\displaystyle @@ -481,7 +1138,7 @@ $$ <<*>>= )clear all ---S 21 of 38 +--S 74 14:491 Axiom cannot compute this integral aa:=integrate(acot(x/a)/x,x) --R --R @@ -502,7 +1159,7 @@ $$ <<*>>= )clear all ---S 22 of 38 +--S 75 aa:=integrate(acot(x/a)/x^2,x) --R --R @@ -514,6 +1171,101 @@ aa:=integrate(acot(x/a)/x^2,x) --R 2a x --R Type: Union(Expression Integer,...) --E + +--S 76 +bb:=-acot(x/a)/x+1/(2*a)*log((x^2+a^2)/x^2) +--R +--R 2 2 +--R x + a x +--R x log(-------) - 2a acot(-) +--R 2 a +--R x +--R (2) --------------------------- +--R 2a x +--R Type: Expression Integer +--E + +--S 77 +cc:=aa-bb +--R +--R (3) +--R 2 2 +--R 2 2 x + a 2a x x +--R x log(x + a ) - 2x log(x) - x log(-------) - a atan(-------) + 2a acot(-) +--R 2 2 2 a +--R x x - a +--R -------------------------------------------------------------------------- +--R 2a x +--R Type: Expression Integer +--E + +--S 78 +acotrule:=rule(acot(x) == -%i/2*log((%i*x-1)/(%i*x+1))) +--R +--R x + %i +--R %i log(------) +--R x - %i +--R (4) acot(x) == - -------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 79 +dd:=acotrule cc +--R +--R (5) +--R 2 2 +--R 2 2 x + %i a x + a +--R x log(x + a ) - 2x log(x) - %i a log(--------) - x log(-------) +--R x - %i a 2 +--R x +--R + +--R 2a x +--R - a atan(-------) +--R 2 2 +--R x - a +--R / +--R 2a x +--R Type: Expression Complex Integer +--E + +--S 80 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (6) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 81 +ee:=atanrule dd +--R +--R (7) +--R 2 2 +--R 2 2 x + 2%i a x - a +--R 2x log(x + a ) - 4x log(x) + %i a log(-----------------) +--R 2 2 +--R x - 2%i a x - a +--R + +--R 2 2 +--R x + %i a x + a +--R - 2%i a log(--------) - 2x log(-------) +--R x - %i a 2 +--R x +--R / +--R 4a x +--R Type: Expression Complex Integer +--E + +--S 82 14:492 Schaums and Axiom agree +ff:=expandLog ee +--R +--R (8) 0 +--R Type: Expression Complex Integer +--E @ \section{\cite{1}:14.493~~~~~$\displaystyle @@ -534,7 +1286,7 @@ $$ <<*>>= )clear all ---S 23 of 38 +--S 83 aa:=integrate(asec(x/a),x) --R --R @@ -555,6 +1307,72 @@ aa:=integrate(asec(x/a),x) --R 2 --R Type: Union(Expression Integer,...) --E + +--S 84 +bb1:=x*asec(x/a)-a*log(x+sqrt(x^2-a^2)) +--R +--R +-------+ +--R | 2 2 x +--R (2) - a log(\|x - a + x) + x asec(-) +--R a +--R Type: Expression Integer +--E + +--S 85 +bb2:=x*asec(x/a)+a*log(x+sqrt(x^2-a^2)) +--R +--R +-------+ +--R | 2 2 x +--R (3) a log(\|x - a + x) + x asec(-) +--R a +--R Type: Expression Integer +--E + +--S 86 +cc1:=aa-bb1 +--R +--R (4) +--R +---------+ +--R +-------+ +-+ | 2 2 +--R | 2 2 +-+ 2x\|2 \|- x + a +--R 2a log(\|x - a + x) - a\|2 atan(------------------) +--R 2 2 +--R 3x - 2a +--R + +--R +---------+ +--R | 2 2 +--R 2a\|- x + a x x +--R x atan(--------------) - 2a atan(------------) - 2x asec(-) +--R 2 +---------+ a +--R x | 2 2 +--R \|- x + a +--R / +--R 2 +--R Type: Expression Integer +--E + +--S 87 14:493 Axiom cannot simplify these expressions +cc2:=aa-bb2 +--R +--R (5) +--R +---------+ +--R +-------+ +-+ | 2 2 +--R | 2 2 +-+ 2x\|2 \|- x + a +--R - 2a log(\|x - a + x) - a\|2 atan(------------------) +--R 2 2 +--R 3x - 2a +--R + +--R +---------+ +--R | 2 2 +--R 2a\|- x + a x x +--R x atan(--------------) - 2a atan(------------) - 2x asec(-) +--R 2 +---------+ a +--R x | 2 2 +--R \|- x + a +--R / +--R 2 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.494~~~~~$\displaystyle @@ -575,7 +1393,7 @@ $$ <<*>>= )clear all ---S 24 of 38 +--S 88 aa:=integrate(x*asec(x/a),x) --R --R @@ -589,6 +1407,60 @@ aa:=integrate(x*asec(x/a),x) --R 4 --R Type: Union(Expression Integer,...) --E + +--S 89 +bb1:=x^2/2*asec(x/a)-(a*sqrt(x^2-a^2))/2 +--R +--R +-------+ +--R | 2 2 2 x +--R - a\|x - a + x asec(-) +--R a +--R (2) ------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 90 +bb2:=x^2/2*asec(x/a)+(a*sqrt(x^2-a^2))/2 +--R +--R +-------+ +--R | 2 2 2 x +--R a\|x - a + x asec(-) +--R a +--R (3) ----------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 91 +cc1:=aa-bb1 +--R +--R (4) +--R +---------+ +--R | 2 2 +-------+ +---------+ +--R 2 2 2a\|- x + a | 2 2 | 2 2 2 x +--R (x - 2a )atan(--------------) + 2a\|x - a + 2a\|- x + a - 2x asec(-) +--R 2 a +--R x +--R --------------------------------------------------------------------------- +--R 4 +--R Type: Expression Integer +--E + +--S 92 14:494 Axiom cannot simplify these expressions +cc2:=aa-bb2 +--R +--R (5) +--R +---------+ +--R | 2 2 +-------+ +---------+ +--R 2 2 2a\|- x + a | 2 2 | 2 2 2 x +--R (x - 2a )atan(--------------) - 2a\|x - a + 2a\|- x + a - 2x asec(-) +--R 2 a +--R x +--R --------------------------------------------------------------------------- +--R 4 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.495~~~~~$\displaystyle @@ -616,7 +1488,7 @@ $$ <<*>>= )clear all ---S 25 of 38 +--S 93 aa:=integrate(x^2*asec(x/a),x) --R --R @@ -638,6 +1510,87 @@ aa:=integrate(x^2*asec(x/a),x) --R 6 --R Type: Union(Expression Integer,...) --E + +--S 94 +bb1:=x^3/3*asec(x/a)-(a*x*sqrt(x^2-a^2))/6-a^3/6*log(x+sqrt(x^2-a^2)) +--R +--R +-------+ +-------+ +--R 3 | 2 2 | 2 2 3 x +--R - a log(\|x - a + x) - a x\|x - a + 2x asec(-) +--R a +--R (2) ---------------------------------------------------- +--R 6 +--R Type: Expression Integer +--E + +--S 95 +bb2:=x^3/3*asec(x/a)+(a*x*sqrt(x^2-a^2))/6+a^3/6*log(x+sqrt(x^2-a^2)) +--R +--R +-------+ +-------+ +--R 3 | 2 2 | 2 2 3 x +--R a log(\|x - a + x) + a x\|x - a + 2x asec(-) +--R a +--R (3) -------------------------------------------------- +--R 6 +--R Type: Expression Integer +--E + +--S 96 +cc1:=aa-bb1 +--R +--R (4) +--R +---------+ +--R +-------+ +-+ | 2 2 +--R 3 | 2 2 3 +-+ 2x\|2 \|- x + a +--R a log(\|x - a + x) - 2a \|2 atan(------------------) +--R 2 2 +--R 3x - 2a +--R + +--R +---------+ +--R | 2 2 +-------+ +--R 3 2a\|- x + a 3 x | 2 2 +--R x atan(--------------) - 5a atan(------------) + a x\|x - a +--R 2 +---------+ +--R x | 2 2 +--R \|- x + a +--R + +--R +---------+ +--R | 2 2 3 x +--R a x\|- x + a - 2x asec(-) +--R a +--R / +--R 6 +--R Type: Expression Integer +--E + +--S 97 14:495 Axiom cannot simplify these expressions +cc2:=aa-bb2 +--R +--R (5) +--R +---------+ +--R +-------+ +-+ | 2 2 +--R 3 | 2 2 3 +-+ 2x\|2 \|- x + a +--R - a log(\|x - a + x) - 2a \|2 atan(------------------) +--R 2 2 +--R 3x - 2a +--R + +--R +---------+ +--R | 2 2 +-------+ +--R 3 2a\|- x + a 3 x | 2 2 +--R x atan(--------------) - 5a atan(------------) - a x\|x - a +--R 2 +---------+ +--R x | 2 2 +--R \|- x + a +--R + +--R +---------+ +--R | 2 2 3 x +--R a x\|- x + a - 2x asec(-) +--R a +--R / +--R 6 +--R Type: Expression Integer +--E + @ \section{\cite{1}:14.496~~~~~$\displaystyle @@ -650,7 +1603,7 @@ $$ <<*>>= )clear all ---S 26 of 38 +--S 98 14:496 Axiom cannot compute this integral aa:=integrate(asec(x/a)/x,x) --R --R @@ -681,7 +1634,7 @@ $$ <<*>>= )clear all ---S 27 of 38 +--S 99 aa:=integrate(asec(x/a)/x^2,x) --R --R @@ -696,6 +1649,70 @@ aa:=integrate(asec(x/a)/x^2,x) --R 2a x\|2 --R Type: Union(Expression Integer,...) --E + +--S 100 +bb1:=-asec(x/a)/x+sqrt(x^2-a^2)/(a*x) +--R +--R +-------+ +--R | 2 2 x +--R \|x - a - a asec(-) +--R a +--R (2) ---------------------- +--R a x +--R Type: Expression Integer +--E + +--S 101 +bb2:=-asec(x/a)/x-sqrt(x^2-a^2)/(a*x) +--R +--R +-------+ +--R | 2 2 x +--R - \|x - a - a asec(-) +--R a +--R (3) ------------------------ +--R a x +--R Type: Expression Integer +--E + +--S 102 +cc1:=aa-bb1 +--R +--R (4) +--R +---------+ +---------+ +--R +-+ | 2 2 | 2 2 +-------+ +--R 2x\|2 \|- x + a +-+ 2a\|- x + a +-+ | 2 2 +--R x atan(------------------) - a\|2 atan(--------------) - 2\|2 \|x - a +--R 2 2 2 +--R 3x - 2a x +--R + +--R +-+ x +--R 2a\|2 asec(-) +--R a +--R / +--R +-+ +--R 2a x\|2 +--R Type: Expression Integer +--E + +--S 103 14:497 Axiom cannot simplify these expressions +cc2:=aa-bb2 +--R +--R (5) +--R +---------+ +---------+ +--R +-+ | 2 2 | 2 2 +-------+ +--R 2x\|2 \|- x + a +-+ 2a\|- x + a +-+ | 2 2 +--R x atan(------------------) - a\|2 atan(--------------) + 2\|2 \|x - a +--R 2 2 2 +--R 3x - 2a x +--R + +--R +-+ x +--R 2a\|2 asec(-) +--R a +--R / +--R +-+ +--R 2a x\|2 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.498~~~~~$\displaystyle @@ -716,7 +1733,7 @@ $$ <<*>>= )clear all ---S 28 of 38 +--S 104 aa:=integrate(acsc(x/a),x) --R --R @@ -737,6 +1754,73 @@ aa:=integrate(acsc(x/a),x) --R 2 --R Type: Union(Expression Integer,...) --E + +--S 105 +bb1:=x*acsc(x/a)+a*log(x+sqrt(x^2-a^2)) +--R +--R +-------+ +--R | 2 2 x +--R (2) a log(\|x - a + x) + x acsc(-) +--R a +--R Type: Expression Integer +--E + +--S 106 +bb2:=x*acsc(x/a)-a*log(x+sqrt(x^2-a^2)) +--R +--R +-------+ +--R | 2 2 x +--R (3) - a log(\|x - a + x) + x acsc(-) +--R a +--R Type: Expression Integer +--E + +--S 107 +cc1:=aa-bb1 +--R +--R (4) +--R +---------+ +--R +-------+ +-+ | 2 2 +--R | 2 2 +-+ 2x\|2 \|- x + a +--R - 2a log(\|x - a + x) + a\|2 atan(------------------) +--R 2 2 +--R 3x - 2a +--R + +--R +---------+ +--R | 2 2 +--R 2a\|- x + a x x +--R - x atan(--------------) + 2a atan(------------) - 2x acsc(-) +--R 2 +---------+ a +--R x | 2 2 +--R \|- x + a +--R / +--R 2 +--R Type: Expression Integer +--E + +--S 108 14:498 Axiom cannot simplify these expressions +cc2:=aa-bb2 +--R +--R (5) +--R +---------+ +--R +-------+ +-+ | 2 2 +--R | 2 2 +-+ 2x\|2 \|- x + a +--R 2a log(\|x - a + x) + a\|2 atan(------------------) +--R 2 2 +--R 3x - 2a +--R + +--R +---------+ +--R | 2 2 +--R 2a\|- x + a x x +--R - x atan(--------------) + 2a atan(------------) - 2x acsc(-) +--R 2 +---------+ a +--R x | 2 2 +--R \|- x + a +--R / +--R 2 +--R Type: Expression Integer +--E + @ \section{\cite{1}:14.499~~~~~$\displaystyle @@ -757,7 +1841,7 @@ $$ <<*>>= )clear all ---S 29 of 38 +--S 109 aa:=integrate(x*acsc(x/a),x) --R --R @@ -771,6 +1855,60 @@ aa:=integrate(x*acsc(x/a),x) --R 4 --R Type: Union(Expression Integer,...) --E + +--S 110 +bb1:=x^2/2*acsc(x/a)+(a*sqrt(x^2-a^2))/2 +--R +--R +-------+ +--R | 2 2 2 x +--R a\|x - a + x acsc(-) +--R a +--R (2) ----------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 111 +bb2:=x^2/2*acsc(x/a)-(a*sqrt(x^2-a^2))/2 +--R +--R +-------+ +--R | 2 2 2 x +--R - a\|x - a + x acsc(-) +--R a +--R (3) ------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 112 +cc1:=aa-bb1 +--R +--R (4) +--R +---------+ +--R | 2 2 +-------+ +---------+ +--R 2 2 2a\|- x + a | 2 2 | 2 2 2 x +--R (- x + 2a )atan(--------------) - 2a\|x - a - 2a\|- x + a - 2x acsc(-) +--R 2 a +--R x +--R ----------------------------------------------------------------------------- +--R 4 +--R Type: Expression Integer +--E + +--S 113 14:499 Axiom cannot simplify these expressions +cc2:=aa-bb2 +--R +--R (5) +--R +---------+ +--R | 2 2 +-------+ +---------+ +--R 2 2 2a\|- x + a | 2 2 | 2 2 2 x +--R (- x + 2a )atan(--------------) + 2a\|x - a - 2a\|- x + a - 2x acsc(-) +--R 2 a +--R x +--R ----------------------------------------------------------------------------- +--R 4 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.500~~~~~$\displaystyle @@ -798,7 +1936,7 @@ $$ <<*>>= )clear all ---S 30 of 38 +--S 114 aa:=integrate(x^2*acsc(x/a),x) --R --R @@ -820,6 +1958,86 @@ aa:=integrate(x^2*acsc(x/a),x) --R 6 --R Type: Union(Expression Integer,...) --E + +--S 115 +bb1:=x^3/3*acsc(x/a)+(a*x*sqrt(x^2-a^2))/6+a^3/6*log(x+sqrt(x^2-a^2)) +--R +--R +-------+ +-------+ +--R 3 | 2 2 | 2 2 3 x +--R a log(\|x - a + x) + a x\|x - a + 2x acsc(-) +--R a +--R (2) -------------------------------------------------- +--R 6 +--R Type: Expression Integer +--E + +--S 116 +bb2:=x^3/3*acsc(x/a)-(a*x*sqrt(x^2-a^2))/6-a^3/6*log(x+sqrt(x^2-a^2)) +--R +--R +-------+ +-------+ +--R 3 | 2 2 | 2 2 3 x +--R - a log(\|x - a + x) - a x\|x - a + 2x acsc(-) +--R a +--R (3) ---------------------------------------------------- +--R 6 +--R Type: Expression Integer +--E + +--S 117 +cc1:=aa-bb1 +--R +--R (4) +--R +---------+ +--R +-------+ +-+ | 2 2 +--R 3 | 2 2 3 +-+ 2x\|2 \|- x + a +--R - a log(\|x - a + x) + 2a \|2 atan(------------------) +--R 2 2 +--R 3x - 2a +--R + +--R +---------+ +--R | 2 2 +-------+ +--R 3 2a\|- x + a 3 x | 2 2 +--R - x atan(--------------) + 5a atan(------------) - a x\|x - a +--R 2 +---------+ +--R x | 2 2 +--R \|- x + a +--R + +--R +---------+ +--R | 2 2 3 x +--R - a x\|- x + a - 2x acsc(-) +--R a +--R / +--R 6 +--R Type: Expression Integer +--E + +--S 118 14:500 Axiom cannot simplify this expression +cc2:=aa-bb2 +--R +--R (5) +--R +---------+ +--R +-------+ +-+ | 2 2 +--R 3 | 2 2 3 +-+ 2x\|2 \|- x + a +--R a log(\|x - a + x) + 2a \|2 atan(------------------) +--R 2 2 +--R 3x - 2a +--R + +--R +---------+ +--R | 2 2 +-------+ +--R 3 2a\|- x + a 3 x | 2 2 +--R - x atan(--------------) + 5a atan(------------) + a x\|x - a +--R 2 +---------+ +--R x | 2 2 +--R \|- x + a +--R + +--R +---------+ +--R | 2 2 3 x +--R - a x\|- x + a - 2x acsc(-) +--R a +--R / +--R 6 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.501~~~~~$\displaystyle @@ -832,7 +2050,7 @@ $$ <<*>>= )clear all ---S 31 of 38 +--S 119 14:501 Axiom cannot compute this integral aa:=integrate(acsc(x/a)/x,x) --R --R @@ -863,7 +2081,7 @@ $$ <<*>>= )clear all ---S 32 of 38 +--S 120 aa:=integrate(acsc(x/a)/x^2,x) --R --R @@ -878,6 +2096,72 @@ aa:=integrate(acsc(x/a)/x^2,x) --R 2a x\|2 --R Type: Union(Expression Integer,...) --E + +--S 121 +bb1:=-acsc(x/a)/x-sqrt(x^2-a^2)/(a*x) +--R +--R +-------+ +--R | 2 2 x +--R - \|x - a - a acsc(-) +--R a +--R (2) ------------------------ +--R a x +--R Type: Expression Integer +--E + +--S 122 +bb2:=-acsc(x/a)/x+sqrt(x^2-a^2)/(a*x) +--R +--R +-------+ +--R | 2 2 x +--R \|x - a - a acsc(-) +--R a +--R (3) ---------------------- +--R a x +--R Type: Expression Integer +--E + +--S 123 +cc1:=aa-bb1 +--R +--R (4) +--R +---------+ +---------+ +--R +-+ | 2 2 | 2 2 +--R 2x\|2 \|- x + a +-+ 2a\|- x + a +--R - x atan(------------------) + a\|2 atan(--------------) +--R 2 2 2 +--R 3x - 2a x +--R + +--R +-------+ +--R +-+ | 2 2 +-+ x +--R 2\|2 \|x - a + 2a\|2 acsc(-) +--R a +--R / +--R +-+ +--R 2a x\|2 +--R Type: Expression Integer +--E + +--S 124 14:502 Axiom cannot simplify this expression +cc2:=aa-bb2 +--R +--R (5) +--R +---------+ +---------+ +--R +-+ | 2 2 | 2 2 +--R 2x\|2 \|- x + a +-+ 2a\|- x + a +--R - x atan(------------------) + a\|2 atan(--------------) +--R 2 2 2 +--R 3x - 2a x +--R + +--R +-------+ +--R +-+ | 2 2 +-+ x +--R - 2\|2 \|x - a + 2a\|2 acsc(-) +--R a +--R / +--R +-+ +--R 2a x\|2 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.503~~~~~$\displaystyle @@ -888,7 +2172,7 @@ $$ <<*>>= )clear all ---S 33 of 38 +--S 125 14:503 Axiom cannot compute this integral aa:=integrate(x^m*asin(x/a),x) --R --R @@ -908,7 +2192,7 @@ $$ <<*>>= )clear all ---S 34 of 38 +--S 126 14:504 Axiom cannot compute this integral aa:=integrate(x^m*acos(x/a),x) --R --R @@ -926,10 +2210,15 @@ $$\int{x^m\tan^{-1}\frac{x}{a}}= \frac{x^{m_1}}{m+1}\tan^{-1}\frac{x}{a} -\frac{a}{m+1}\int{\frac{x^{m+1}}{x^2+a^2}} $$ +This appears to be an interesting integral. Axiom found a closed +form solution to the problem. However, the t1 integral below does +not have a closed form solution. Note that we did not return a +result for the prior two integrals, nor for the next integral. They +have the same form but are expressed in terms of asin, acos, and acot. <<*>>= )clear all ---S 35 of 38 +--S 127 aa:=integrate(x*m*atan(x/a),x) --R --R @@ -941,6 +2230,97 @@ aa:=integrate(x*m*atan(x/a),x) --R 4 --R Type: Union(Expression Integer,...) --E + +--S 128 +t1:=integrate(x^(m+1)/(x^2+a^2),x) +--E +@ +Since we cannot get a closed form version of the prior integral +we proceed to try to prove that Axiom got a correct answer. We +do this by computing the derivate of 'aa' above and finding the +difference from the original formula. + +So first we generate the derivative: +<<*>>= + +--S 129 +bb:=D(aa,x) +--R +--R 2a x +--R m x atan(-------) +--R 2 2 +--R x - a +--R (3) - ----------------- +--R 2 +--R Type: Expression Integer +--E +@ +Then we input the original expression +<<*>>= +--S 130 +aa1:=x*m*atan(x/a) +--R +--R x +--R (4) m x atan(-) +--R a +--R Type: Expression Integer +--E +@ +Now we take their difference +<<*>>= +--S 131 +dd:=aa1-bb +--R +--R x 2a x +--R 2m x atan(-) + m x atan(-------) +--R a 2 2 +--R x - a +--R (5) -------------------------------- +--R 2 +--R Type: Expression Integer +--E +@ +Now we input the atan transformation +<<*>>= +--S 132 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (6) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E +@ +And apply the transformation to the difference +<<*>>= +--S 133 +ee:=atanrule dd +--R +--R 2 2 +--R x + 2%i a x - a - x + %i a +--R - %i m x log(-----------------) - 2%i m x log(----------) +--R 2 2 x + %i a +--R x - 2%i a x - a +--R (7) --------------------------------------------------------- +--R 4 +--R Type: Expression Complex Integer +--E +@ +And now we simplify +<<*>>= +--S 134 14:505 SCHAUMS AND AXIOM DISAGREE? (branch cuts?) +ff:=expandLog ee +--R +--R %i m x log(- 1) +--R (8) - --------------- +--R 2 +--R Type: Expression Complex Integer +--E +And we get the surprising result that they are not equal. +In fact, they differ by a complex value depending on x. +Likely there is a branch-cut issue lurking somewhere. @ \section{\cite{1}:14.506~~~~~$\displaystyle @@ -952,7 +2332,7 @@ $$ <<*>>= )clear all ---S 36 of 38 +--S 135 14:506 Axiom cannot compute this integral aa:=integrate(x^m*acot(x/a),x) --R --R @@ -982,7 +2362,7 @@ $$ <<*>>= )clear all ---S 37 of 38 +--S 136 14:507 Axiom cannot compute this integral aa:=integrate(x^m*asec(x/a),x) --R --R @@ -1012,7 +2392,7 @@ $$ <<*>>= )clear all ---S 38 of 38 +--S 137 14:508 Axiom cannot compute this integral aa:=integrate(x^m*acsc(x/a),x) --R --R diff --git a/src/input/schaum25.input.pamphlet b/src/input/schaum25.input.pamphlet index 250e7c3..0a55f34 100644 --- a/src/input/schaum25.input.pamphlet +++ b/src/input/schaum25.input.pamphlet @@ -18,16 +18,32 @@ $$ )set message auto off )clear all ---S 1 of 16 -aa:=integrate(e^(a*x),x) ---R +--S 1 +aa:=integrate(%e^(a*x),x) --R ---R a x log(e) +--R a x --R %e ---R (1) ------------ ---R a log(e) +--R (1) ----- +--R a --R Type: Union(Expression Integer,...) --E + +--S 2 +bb:=%e^(a*x)/a +--R +--R a x +--R %e +--R (2) ----- +--R a +--R Type: Expression Integer +--E + +--S 3 14:509 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.510~~~~~$\displaystyle @@ -38,17 +54,34 @@ $$ <<*>>= )clear all ---S 2 of 16 -aa:=integrate(x*e^(a*x),x) ---R +--S 4 +aa:=integrate(x*%e^(a*x),x) --R ---R a x log(e) ---R (a x log(e) - 1)%e ---R (1) ---------------------------- ---R 2 2 ---R a log(e) +--R a x +--R (a x - 1)%e +--R (1) -------------- +--R 2 +--R a --R Type: Union(Expression Integer,...) --E + +--S 5 +bb:=%e^(a*x)/a*(x-1/a) +--R +--R a x +--R (a x - 1)%e +--R (2) -------------- +--R 2 +--R a +--R Type: Expression Integer +--E + +--S 6 14:510 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.511~~~~~$\displaystyle @@ -59,17 +92,34 @@ $$ <<*>>= )clear all ---S 3 of 16 -aa:=integrate(x^2*e^(a*x),x) ---R +--S 7 +aa:=integrate(x^2*%e^(a*x),x) --R ---R 2 2 2 a x log(e) ---R (a x log(e) - 2a x log(e) + 2)%e ---R (1) ------------------------------------------- ---R 3 3 ---R a log(e) +--R 2 2 a x +--R (a x - 2a x + 2)%e +--R (1) ---------------------- +--R 3 +--R a --R Type: Union(Expression Integer,...) --E + +--S 8 +bb:=%e^(a*x)/a*(x^2-(2*x)/a+2/a^2) +--R +--R 2 2 a x +--R (a x - 2a x + 2)%e +--R (2) ---------------------- +--R 3 +--R a +--R Type: Expression Integer +--E + +--S 9 14:511 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.512~~~~~$\displaystyle @@ -89,16 +139,16 @@ $$ <<*>>= )clear all ---S 4 of 16 -aa:=integrate(x^n*e^(a*x),x) ---R +--S 10 14:512 Axiom cannot compute this integral +aa:=integrate(x^n*%e^(a*x),x) --R --R x ---I ++ n %I a ---I (1) | %I e d%I +--I ++ %I a n +--I (1) | %e %I d%I --R ++ --R Type: Union(Expression Integer,...) ---E +--E + @ \section{\cite{1}:14.513~~~~~$\displaystyle @@ -110,11 +160,10 @@ $$ <<*>>= )clear all ---S 5 of 16 -aa:=integrate(e^(a*x)/x,x) ---R +--S 11 14:513 Schaums and Axiom agree by definition +aa:=integrate(%e^(a*x)/x,x) --R ---R (1) Ei(a x log(e)) +--R (1) Ei(a x) --R Type: Union(Expression Integer,...) --E @ @@ -127,15 +176,14 @@ $$ <<*>>= )clear all ---S 6 of 16 -aa:=integrate(e^(a*x)/x^n,x) ---R +--S 12 14:514 Axiom cannot compute this integral +aa:=integrate(%e^(a*x)/x^n,x) --R ---I x %I a ---R ++ e ---I (1) | ----- d%I ---R ++ n ---I %I +--I x %I a +--R ++ %e +--I (1) | ------ d%I +--R ++ n +--I %I --R Type: Union(Expression Integer,...) --E @ @@ -148,16 +196,32 @@ $$ <<*>>= )clear all ---S 7 of 16 -aa:=integrate(1/(p+q*e^(a*x)),x) ---R +--S 13 +aa:=integrate(1/(p+q*%e^(a*x)),x) --R ---R a x log(e) ---R - log(q %e + p) + a x log(e) ---R (1) -------------------------------------- ---R a p log(e) +--R a x +--R - log(q %e + p) + a x +--R (1) ------------------------ +--R a p --R Type: Union(Expression Integer,...) --E + +--S 14 +bb:=x/p-1/(a*p)*log(p+q*%e^(a*x)) +--R +--R a x +--R - log(q %e + p) + a x +--R (2) ------------------------ +--R a p +--R Type: Expression Integer +--E + +--S 15 14:515 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.516~~~~~$\displaystyle @@ -168,16 +232,34 @@ $$ <<*>>= )clear all ---S 8 of 16 -aa:=integrate(1/(p+q*e^(a*x)),x) ---R +--S 16 +aa:=integrate(1/(p+q*%e^(a*x))^2,x) --R ---R a x log(e) ---R - log(q %e + p) + a x log(e) ---R (1) -------------------------------------- ---R a p log(e) +--R a x a x a x +--R (- q %e - p)log(q %e + p) + a q x %e + a p x + p +--R (1) --------------------------------------------------------- +--R 2 a x 3 +--R a p q %e + a p --R Type: Union(Expression Integer,...) --E + +--S 17 +bb:=x/p^2+1/(a*p*(p+q*%e^(a*x)))-1/(a*p^2)*log(p+q*%e^(a*x)) +--R +--R a x a x a x +--R (- q %e - p)log(q %e + p) + a q x %e + a p x + p +--R (2) --------------------------------------------------------- +--R 2 a x 3 +--R a p q %e + a p +--R Type: Expression Integer +--E + +--S 18 14:516 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.517~~~~~$\displaystyle @@ -197,16 +279,114 @@ $$ <<*>>= )clear all ---S 9 of 16 -aa:=integrate(1/(p*e^(a*x)+q*e^(a*x)),x) ---R +--S 19 +aa:=integrate(1/(p*%e^(a*x)+q*%e^-(a*x)),x) --R ---R 1 ---R (1) - ----------------------------- ---R a x log(e) ---R (a q + a p)log(e)%e ---R Type: Union(Expression Integer,...) +--R a x 2 +-----+ a x +--R (p (%e ) - q)\|- p q + 2p q %e a x +---+ +--R log(-------------------------------------) %e \|p q +--R a x 2 atan(-----------) +--R p (%e ) + q q +--R (1) [------------------------------------------,-----------------] +--R +-----+ +---+ +--R 2a\|- p q a\|p q +--R Type: Union(List Expression Integer,...) --E + +--S 20 +bb1:=1/(a*sqrt(p*q))*atan(sqrt(p/q)*%e^(a*x)) +--R +--R +-+ +--R a x |p +--R atan(%e |- ) +--R \|q +--R (2) --------------- +--R +---+ +--R a\|p q +--R Type: Expression Integer +--E + +--S 21 +bb2:=1/(2*a*sqrt(-p*q))*log((%e^(a*x)-sqrt(-q/p))/(%e^(a*x)+sqrt(-q/p))) +--R +--R +---+ +--R | q a x +--R - |- - + %e +--R \| p +--R log(----------------) +--R +---+ +--R | q a x +--R |- - + %e +--R \| p +--R (3) --------------------- +--R +-----+ +--R 2a\|- p q +--R Type: Expression Integer +--E + +--S 22 +cc1:=aa.1-bb1 +--R +--R (4) +--R a x 2 +-----+ a x +-+ +--R +---+ (p (%e ) - q)\|- p q + 2p q %e +-----+ a x |p +--R \|p q log(-------------------------------------) - 2\|- p q atan(%e |- ) +--R a x 2 \|q +--R p (%e ) + q +--R --------------------------------------------------------------------------- +--R +-----+ +---+ +--R 2a\|- p q \|p q +--R Type: Expression Integer +--E + +--S 23 +cc2:=aa.2-bb1 +--R +--R a x +---+ +-+ +--R %e \|p q a x |p +--R atan(-----------) - atan(%e |- ) +--R q \|q +--R (5) ----------------------------------- +--R +---+ +--R a\|p q +--R Type: Expression Integer +--E + +--S 24 +cc3:=aa.1-bb2 +--R +--R +---+ +--R | q a x +--R a x 2 +-----+ a x - |- - + %e +--R (p (%e ) - q)\|- p q + 2p q %e \| p +--R log(-------------------------------------) - log(----------------) +--R a x 2 +---+ +--R p (%e ) + q | q a x +--R |- - + %e +--R \| p +--R (6) ------------------------------------------------------------------ +--R +-----+ +--R 2a\|- p q +--R Type: Expression Integer +--E + +--S 25 14:517 Axiom cannot simplify these expressions +cc4:=aa.2-bb2 +--R +--R +---+ +--R | q a x +--R - |- - + %e a x +---+ +--R +---+ \| p +-----+ %e \|p q +--R - \|p q log(----------------) + 2\|- p q atan(-----------) +--R +---+ q +--R | q a x +--R |- - + %e +--R \| p +--R (7) ---------------------------------------------------------- +--R +-----+ +---+ +--R 2a\|- p q \|p q +--R Type: Expression Integer +--E @ \section{\cite{1}:14.518~~~~~$\displaystyle @@ -217,17 +397,34 @@ $$ <<*>>= )clear all ---S 10 of 16 -aa:=integrate(e^(a*x)*sin(b*x),x) ---R +--S 26 +aa:=integrate(%e^(a*x)*sin(b*x),x) --R ---R a x log(e) ---R (a log(e)sin(b x) - b cos(b x))%e ---R (1) ------------------------------------------- ---R 2 2 2 ---R a log(e) + b +--R a x a x +--R a %e sin(b x) - b cos(b x)%e +--R (1) --------------------------------- +--R 2 2 +--R b + a --R Type: Union(Expression Integer,...) --E + +--S 27 +bb:=((%e^(a*x))*(a*sin(b*x)-b*cos(b*x)))/(a^2+b^2) +--R +--R a x a x +--R a %e sin(b x) - b cos(b x)%e +--R (2) --------------------------------- +--R 2 2 +--R b + a +--R Type: Expression Integer +--E + +--S 28 14:518 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.519~~~~~$\displaystyle @@ -238,17 +435,34 @@ $$ <<*>>= )clear all ---S 11 of 16 -aa:=integrate(e^(a*x)*cos(b*x),x) ---R +--S 29 +aa:=integrate(%e^(a*x)*cos(b*x),x) --R ---R a x log(e) ---R (b sin(b x) + a cos(b x)log(e))%e ---R (1) ------------------------------------------- ---R 2 2 2 ---R a log(e) + b +--R a x a x +--R b %e sin(b x) + a cos(b x)%e +--R (1) --------------------------------- +--R 2 2 +--R b + a --R Type: Union(Expression Integer,...) --E + +--S 30 +bb:=((%e^(a*x))*(a*cos(b*x)+b*sin(b*x)))/(a^2+b^2) +--R +--R a x a x +--R b %e sin(b x) + a cos(b x)%e +--R (2) --------------------------------- +--R 2 2 +--R b + a +--R Type: Expression Integer +--E + +--S 31 14:519 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.520~~~~~$\displaystyle @@ -260,24 +474,36 @@ $$ <<*>>= )clear all ---S 12 of 16 -aa:=integrate(x*e^(a*x)*sin(b*x),x) ---R +--S 32 +aa:=integrate(x*%e^(a*x)*sin(b*x),x) --R --R (1) ---R 3 3 2 2 2 2 ---R (a x log(e) - a log(e) + a b x log(e) + b )sin(b x) ---R + ---R 2 2 3 ---R - a b x cos(b x)log(e) + 2a b cos(b x)log(e) - b x cos(b x) ---R * ---R a x log(e) ---R %e ---R / ---R 4 4 2 2 2 4 ---R a log(e) + 2a b log(e) + b +--R 2 3 2 2 a x 3 2 a x +--R ((a b + a )x + b - a )%e sin(b x) + ((- b - a b)x + 2a b)cos(b x)%e +--R --------------------------------------------------------------------------- +--R 4 2 2 4 +--R b + 2a b + a --R Type: Union(Expression Integer,...) --E + +--S 33 +bb:=(x*%e^(a*x)*(a*sin(b*x)-b*cos(b*x)))/(a^2+b^2)-(%e^(a*x)*((a^2-b^2)*sin(b*x)-2*a*b*cos(b*x)))/(a^2+b^2)^2 +--R +--R (2) +--R 2 3 2 2 a x 3 2 a x +--R ((a b + a )x + b - a )%e sin(b x) + ((- b - a b)x + 2a b)cos(b x)%e +--R --------------------------------------------------------------------------- +--R 4 2 2 4 +--R b + 2a b + a +--R Type: Expression Integer +--E + +--S 34 14:520 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.521~~~~~$\displaystyle @@ -289,24 +515,36 @@ $$ <<*>>= )clear all ---S 13 of 16 -aa:=integrate(x*e^(a*x)*cos(b*x),x) ---R +--S 35 +aa:=integrate(x*%e^(a*x)*cos(b*x),x) --R --R (1) ---R 2 2 3 3 3 ---R (a b x log(e) - 2a b log(e) + b x)sin(b x) + a x cos(b x)log(e) ---R + ---R 2 2 2 2 ---R - a cos(b x)log(e) + a b x cos(b x)log(e) + b cos(b x) ---R * ---R a x log(e) ---R %e ---R / ---R 4 4 2 2 2 4 ---R a log(e) + 2a b log(e) + b +--R 3 2 a x 2 3 2 2 a x +--R ((b + a b)x - 2a b)%e sin(b x) + ((a b + a )x + b - a )cos(b x)%e +--R ------------------------------------------------------------------------- +--R 4 2 2 4 +--R b + 2a b + a --R Type: Union(Expression Integer,...) --E + +--S 36 +bb:=(x*%e^(a*x)*(a*cos(b*x)+b*sin(b*x)))/(a^2+b^2)-(%e^(a*x)*((a^2-b^2)*cos(b*x)+2*a*b*sin(b*x)))/(a^2+b^2)^2 +--R +--R (2) +--R 3 2 a x 2 3 2 2 a x +--R ((b + a b)x - 2a b)%e sin(b x) + ((a b + a )x + b - a )cos(b x)%e +--R ------------------------------------------------------------------------- +--R 4 2 2 4 +--R b + 2a b + a +--R Type: Expression Integer +--E + +--S 37 14:521 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.522~~~~~$\displaystyle @@ -317,14 +555,13 @@ $$ <<*>>= )clear all ---S 14 of 16 -aa:=integrate(e^(a*x)*log(x),x) ---R +--S 38 14:522 Schaums and Axiom agree by definition +aa:=integrate(%e^(a*x)*log(x),x) --R ---R x ---I ++ %I a ---I (1) | log(%I)e d%I ---R ++ +--R a x +--R %e log(x) - Ei(a x) +--R (1) --------------------- +--R a --R Type: Union(Expression Integer,...) --E @ @@ -338,13 +575,12 @@ $$ <<*>>= )clear all ---S 15 of 16 -aa:=integrate(e^(a*x)*sin(b*x)^n,x) ---R +--S 39 14:523 Axiom cannot compute this integral +aa:=integrate(%e^(a*x)*sin(b*x)^n,x) --R --R x ---I ++ %I a n ---I (1) | e sin(%I b) d%I +--I ++ %I a n +--I (1) | %e sin(%I b) d%I --R ++ --R Type: Union(Expression Integer,...) --E @@ -359,13 +595,12 @@ $$ <<*>>= )clear all ---S 16 of 16 -aa:=integrate(e^(a*x)*cos(b*x)^n,x) ---R +--S 40 14:524 Axiom cannot compute this integral +aa:=integrate(%e^(a*x)*cos(b*x)^n,x) --R --R x ---I ++ %I a n ---I (1) | e cos(%I b) d%I +--I ++ %I a n +--I (1) | %e cos(%I b) d%I --R ++ --R Type: Union(Expression Integer,...) --E