#1: " ----------[ D e r i v e ]---------- " User #2: " ---------- Initialization ---------- " User #3: " ---------- Indefinite Integrals ---------- " User #4: " This integral only makes sense for x real => x |x|/2 "User / #5: / |x| dx User x*|x| #6: ------- Simp(#5) 2 #7: z :epsilon Complex User #8: z Simp(#7) / #9: / |z| dz User z*|z| #10: ------- Simp(#9) 2 #11: z := User #12: " Calculus on a piecewise defined function " User #13: A(x) := IF(x < 0, -x, x) User #14: " => if x < 0 then -x^2/2 else x^2/2 " User / #15: / A(x) dx User / #16: / IF(x < 0, -x, x) dx Simp(#15) #17: A(x) := - x*CHI(-inf, x, 0) + x*CHI(0, x, inf) User / #18: / A(x) dx User x*|x| #19: ------- Simp(#18) 2 #20: a := User #21: " This would be very difficult to do by hand " User User #22: " => 2^(1/3)/6 [1/2 log([x + 2^(1/3)]^2/[x^2 - 2^(1/3) x + 2^(2/3)]) " User #23: " + sqrt(3) arctan({[sqrt(3) x]/[2^(4/3) - x] or " User #24: " [2 x - 2^(1/3)]/[2^(1/3) sqrt(3)]}) " #25: " [Gradshteyn and Ryzhik 2.126(1)] " User 1 q1_ := -------- #26: 3 User x + 2 / #27: q2_ := / q1_ dx User #28: q2_ User Simp(#28) / 2/3 \ 1/6 | SQRT(3)*(2 *x - 1) | 108 *ATAN|----------------------| #29: \ 3 / ------------------------------------- - 6 1/3 2 1/3 2/3 1/3 1/3 2 *LN(x - 2 *x + 2 ) 2 *LN(x + 2 ) ----------------------------- + ------------------- 12 6 d #30: q3_ := -- q2_ User dx #31: q3_ User 1 --------------------------------- #32: 1/3 2 1/3 2/3 Simp(#31) (x + 2 )*(x - 2 *x + 2 ) 1 --------------- #33: / 1 \ User EXPAND|-----| \ q3_ / 1 -------- #34: 3 Simp(#33) x + 2 User #35: " This integral is easy if one realizes that 4^x = (2^x)^2 " User #36: " => arcsinh(2^x)/log(2) [Robert Israel in sci.math.symbolic] " / x | 2 #37: | -------------- dx User | x / SQRT(1 + 4 ) 2*x x LN(SQRT(2 + 1) + 2 ) #38: ------------------------- Simp(#37) LN(2) #39: " => (-9 x^2 + 16 x - 41/5)/(2 x - 1)^(5/2) " User #40: " [Gradshteyn and Ryzhik 2.244(8)] " User / 2 | (3*x - 5) #41: | -------------- dx User | 7/2 / (2*x - 1) 2 45*x - 80*x + 41 #42: - ------------------- Simp(#41) 5/2 5*(2*x - 1) User #43: " => 1/[2 m sqrt(10)] log([-5 + e^(m x) sqrt(10)]/[-5 - e^(m x) sqrt(10)]) " #44: " [Gradshteyn and Ryzhik 2.314] " User / 1 #45: | --------------------------- dx User / 2*EXP(m*x) - 5*EXP(- m*x) / m*x \ | SQRT(2)*#e - SQRT(5) | SQRT(10)*LN|-------------------------| #46: | m*x | Simp(#45) \ SQRT(2)*#e + SQRT(5) / ---------------------------------------- 20*m User #47: " => -3/2 x + 1/4 sinh(2 x) + tanh x [Gradshteyn and Ryzhik 2.423(24)] " / 4 | SINH(x) #48: | ---------- dx User | 2 / COSH(x) Simp(#48) - 2*x 6*x 4*x 2*x #e *(#e + #e *(1 - 12*x) - #e *(12*x + 17) - 1) #49: ------------------------------------------------------------ 2*x 8*(#e + 1) #50: " This example involves several symbolic parameters " User User #51: " => 1/sqrt(b^2 - a^2) log([sqrt(b^2 - a^2) tan(x/2) + a + b]/ " User #52: " [sqrt(b^2 - a^2) tan(x/2) - a - b]) (a^2 < b^2) " #53: " [Gradshteyn and Ryzhik 2.553(3)] " User / 1 #54: q_ := | -------------- dx User / a + b*COS(x) #55: q_ User Simp(#55) / 2*b*SIN(x) \ x - 2*ATAN|-------------------------------------------| | 2 | #56: \ 2*b*COS(x) + (SQRT(a - b) + SQRT(a + b)) / --------------------------------------------------------- SQRT(a + b)*SQRT(a - b) d #57: -- q_ User dx 1 #58: -------------- Simp(#57) b*COS(x) + a User #59: " The integral of 1/(a + 3 cos x + 4 sin x) can have 4 different forms " User #60: " depending on the value of a ! [Gradshteyn and Ryzhik 2.558(4)] " #61: " => (a = 3) 1/4 log[3 + 4 tan(x/2)] " User / 1 #62: | ------------------------- dx User / 3 + 3*COS(x) + 4*SIN(x) Simp(#62) / 3*COS(x) + 4*SIN(x) + 3 \ / x 1 \ LN|-------------------------| pi*#i*FLOOR|------ + ---| #63: \ COS(x) + 1 / \ 2*pi 2 / ------------------------------- - --------------------------- 4 4 #64: " => (a = 4) 1/3 log([tan(x/2) + 1]/[tan(x/2) + 7]) " User / 1 #65: | ------------------------- dx User / 4 + 3*COS(x) + 4*SIN(x) Simp(#65) LN(COS(x) + SIN(x) + 1) LN(7*COS(x) + SIN(x) + 7) #66: ------------------------- - --------------------------- 3 3 #67: " => (a = 5) -1/[2 + tan(x/2)] " User / 1 #68: | ------------------------- dx User / 5 + 3*COS(x) + 4*SIN(x) COS(x) + 1 #69: - ----------------------- Simp(#68) 2*COS(x) + SIN(x) + 2 User #70: " => (a = 6) 2/sqrt(11) arctan([3 tan(x/2) + 4]/sqrt(11)) " / 1 #71: | ------------------------- dx User / 6 + 3*COS(x) + 4*SIN(x) Simp(#71) / 4*COS(x) - 3*SIN(x) \ 2*SQRT(11)*ATAN|------------------------------------| #72: \ 3*COS(x) + 4*SIN(x) + SQRT(11) + 6 / ------------------------------------------------------- + 11 SQRT(11)*x ------------ 11 #73: " => x log|x^2 - a^2| - 2 x + a log|(x + a)/(x - a)| " User #74: " [Gradshteyn and Ryzhik 2.736(1)] " User / | 2 2| #75: / LOG(|x - a |) dx User Simp(#75) / x - a \ 2 2 / #76: - a*LN|-------| + x*LN(x - a ) - 2*x + #i*|SIGN(x - \ x + a / \ / pi*|x + a| \ pi*x \ a)*|------------ - pi*|a|| - ------| \ 2 / 2 / User #77: " => (a x)/2 + (pi x^2)/4 - 1/2 (x^2 + a^2) arctan(x/a) " #78: " [Gradshteyn and Ryzhik 2.822(4)] or " User User #79: " (a x)/2 + 1/2 (x^2 + a^2) arccot(x/a) [Gradshteyn and Ryzhik 2.853(2)] " / / x \ #80: | x*ACOT|---| dx User / \ a / 2 2 / x \ (x + a )*ATAN|---| 2 #81: \ a / pi*x a*x Simp(#80) - --------------------- + ------- + ----- 2 4 2 #82: " => [sin(5 x) Ci(2 x)]/5 - [Si(7 x) + Si(3 x)]/10 " User #83: " [Gradshteyn and Ryzhik 5.31(1)] " User / #84: / COS(5*x)*CI(2*x) dx User Simp(#84) / | | #85: | SIN(5*x)*LN(2*x) | euler_gamma ------------------ + SIN(5*x)*|------------- + 5 \ 5 2*x \ / / COS(t_) 1 \ | | |--------- - ----| dt_ | / COS(2*x)*SIN(5*x) / \ t_ t_ / | | ------------------- dx 0 | / x -----------------------------| - -------------------------- 5 / 5 User #86: " => 1/2 [f(x) - g(x)]/[f(x) + g(x)] [Gradshteyn and Ryzhik 2.02(25)] " #87: F(x) := User #88: G(x) := User / /d \ d | |-- F(x)|*G(x) - F(x)*-- G(x) | \dx / dx #89: | ------------------------------- dx User | 2 2 / F(x) - G(x) Simp(#89) / F'(x) / F'(x) | ------------- dx | ------------- dx #90: / F(x) - G(x) / F(x) + G(x) -------------------- - -------------------- - 2 2 / G'(x) / G'(x) | ------------- dx + | ------------- dx / F(x) - G(x) / F(x) + G(x) ----------------------------------------- 2 #91: f := User #92: g := User #93: " ---------- Quit ---------- " User