求不定积分∫x/(1-x^3)dx=

Question

求不定积分∫x/(1-x^3)dx=

Answer 1

x/(1 - x³) = x/[(1 - x)(1 + x + x²)] = A/(1 - x) + (Bx + C)/(1 + x + x²)
x = A(1 + x + x²) + (Bx + C)(1 - x)
x = (A - B)x² + (A + B - C)x + (A + C)
A - B = 0 ==> B = A
A + B - C = 1
A + C = 0 ==> C = - A
A + B - C = 1
A + A + A = 1 ==> A = 1/3
B = 1/3,C = - 1/3
x/(1 - x³) = (x - 1)/[3(x² + x + 1)] - 1/[3(x - 1)]
∫ x/(1 - x³) dx
= (1/3)∫ (x - 1)/(x² + x + 1) dx - (1/3)∫ 1/(x - 1) dx
= (1/3)∫ [(1/2)(2x + 1 - 1) - 1]/(x² + x + 1) dx - (1/3)∫ 1/(x - 1) d(x - 1)
= (1/6)∫ (2x + 1)/(x² + x + 1) dx - (1/2)∫ 1/(x² + x + 1) dx - (1/3)Ln|x - 1|
= (1/6)∫ d(x² + x + 1)/(x² + x + 1) - (1/2)∫ 1/[(x + 1/2)² + 3/4] - (1/3)Ln|x - 1|
= (1/6)Ln|x² + x + 1| - (1/2)(2/√3)arctan[(x + 1/2) • 2/√3] - (1/3)Ln|x - 1| + C
= (1/6)Ln(x² + x + 1) - (1/3)Ln|x - 1| - (1/√3)arctan[(2x + 1)/√3] + C