3/(x^3 + 1)
= 3/[(x+1)(x^2 - x + 1)]
= 3[a/(x+1) + (bx+c)/(x^2 - x + 1)]
a(x^2 - x + 1) + (bx+c)(x+1)
= ax^2 - ax + a + bx^2 + cx + bx + c
= (a+b)x^2 + (b+c-a)x + (a+c)
= 1
a + b = 0,
b + c - a = 0,
a + c = 1,
a = 1/3,b = -1/3,c = 2/3.
3/(x^3 + 1)
= 3[a/(x+1) + (bx+c)/(x^2 - x + 1)]
= 1/(x+1) + (2-x)/(x^2 - x + 1)
= 1/(x+1) + (-1/2)(2x-1)/(x^2 - x + 1) + (3/2)/(x^2 - x + 1/4 + 3/4)
= 1/(x+1) + (-1/2)(2x-1)/(x^2 - x + 1) + (3/2)/[(x - 1/2)^2 + 3/4]
= 1/(x+1) + (-1/2)(2x-1)/(x^2 - x + 1) + 3^(1/2)[2/3^(1/2)]/{[2/3^(1/2)(x - 1/2)]^2 + 1}
积分为,
ln|x+1| - 1/2ln(x^2 - x + 1) + 3^(1/2)arctan[(2x-1)/3^(1/2)] + C
(x^2 + 1)dx/(x^4 + 1)
= (1 + 1/x^2)dx/(x^2 + 1/x^2)
= d(x - 1/x)/[(x - 1/x)^2 + 2]
= 2^(-1/2)d[(x - 1/x)/2^(1/2)]/{[(x - 1/x)/2^(1/2)]^2 + 1}
积分为,
2^(-1/2)arctan[(x - 1/x)/2^(1/2)] + C,
其中,C为任意常数.
