1 / | | a*x | E *sin(b*x) dx | / 0
Integral(E^(a*x)*sin(b*x), (x, 0, 1))
// 0 for And(a = 0, b = 0)\ || | || -I*b*x -I*b*x -I*b*x | ||x*e *sin(b*x) I*e *sin(b*x) I*x*cos(b*x)*e | ||------------------ + ------------------ - -------------------- for a = -I*b | / || 2 2*b 2 | | || | | a*x || I*b*x I*b*x I*b*x | | E *sin(b*x) dx = C + |< x*e *sin(b*x) I*x*cos(b*x)*e I*e *sin(b*x) | | || ----------------- + ------------------- - ----------------- for a = I*b | / || 2 2 2*b | || | || a*x a*x | || a*e *sin(b*x) b*cos(b*x)*e | || --------------- - --------------- otherwise | || 2 2 2 2 | \\ a + b a + b /
/ 0 for Or(And(a = 0, b = 0), And(a = 0, a = -I*b, b = 0), And(a = 0, a = I*b, b = 0), And(a = 0, a = -I*b, a = I*b, b = 0)) | | -I*b -I*b -I*b | 1 e *sin(b) I*cos(b)*e cos(b)*e |--- + ------------ - -------------- - ------------ for Or(And(a = 0, a = -I*b), And(a = -I*b, a = I*b), And(a = -I*b, b = 0), And(a = 0, a = -I*b, a = I*b), And(a = -I*b, a = I*b, b = 0), a = -I*b) |2*b 2 2 2*b | | I*b I*b I*b < 1 e *sin(b) I*cos(b)*e cos(b)*e | --- + ----------- + ------------- - ----------- for Or(And(a = 0, a = I*b), And(a = I*b, b = 0), a = I*b) | 2*b 2 2 2*b | | a a | b a*e *sin(b) b*cos(b)*e | ------- + ----------- - ----------- otherwise | 2 2 2 2 2 2 \ a + b a + b a + b
=
/ 0 for Or(And(a = 0, b = 0), And(a = 0, a = -I*b, b = 0), And(a = 0, a = I*b, b = 0), And(a = 0, a = -I*b, a = I*b, b = 0)) | | -I*b -I*b -I*b | 1 e *sin(b) I*cos(b)*e cos(b)*e |--- + ------------ - -------------- - ------------ for Or(And(a = 0, a = -I*b), And(a = -I*b, a = I*b), And(a = -I*b, b = 0), And(a = 0, a = -I*b, a = I*b), And(a = -I*b, a = I*b, b = 0), a = -I*b) |2*b 2 2 2*b | | I*b I*b I*b < 1 e *sin(b) I*cos(b)*e cos(b)*e | --- + ----------- + ------------- - ----------- for Or(And(a = 0, a = I*b), And(a = I*b, b = 0), a = I*b) | 2*b 2 2 2*b | | a a | b a*e *sin(b) b*cos(b)*e | ------- + ----------- - ----------- otherwise | 2 2 2 2 2 2 \ a + b a + b a + b
Piecewise((0, ((a = 0)∧(b = 0))∨((a = 0)∧(b = 0)∧(a = i*b))∨((a = 0)∧(b = 0)∧(a = -i*b))∨((a = 0)∧(b = 0)∧(a = i*b)∧(a = -i*b))), (1/(2*b) + exp(-i*b)*sin(b)/2 - i*cos(b)*exp(-i*b)/2 - cos(b)*exp(-i*b)/(2*b), (a = -i*b)∨((a = 0)∧(a = -i*b))∨((b = 0)∧(a = -i*b))∨((a = i*b)∧(a = -i*b))∨((a = 0)∧(a = i*b)∧(a = -i*b))∨((b = 0)∧(a = i*b)∧(a = -i*b))), (1/(2*b) + exp(i*b)*sin(b)/2 + i*cos(b)*exp(i*b)/2 - cos(b)*exp(i*b)/(2*b), (a = i*b)∨((a = 0)∧(a = i*b))∨((b = 0)∧(a = i*b))), (b/(a^2 + b^2) + a*exp(a)*sin(b)/(a^2 + b^2) - b*cos(b)*exp(a)/(a^2 + b^2), True))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.