求定积分∫（0,1）(X^3-X)/(X^2+1)^3DX
问题描述：
求定积分∫（0,1）(X^3-X)/(X^2+1)^3DX
答
令 u = x², du = 2x dx
I = (1/2) ∫[0,1] (u-1)/(u+1)³ du 令 v = u+1
= (1/2) ∫ [1,2] (v - 2) / v³ dv
= (1/2) ( -1/v + 1/v² ) | [1,2]
= (1/2) ( 1/2 - 3/4) = - 1/8
答
∫ (0-->1) (x^3-x)/(x^2+1)^3 dx
分子分母同除以x^3
=∫ (0-->1) (1-1/x^2)/(x+1/x)^3 dx
分子放到微分d的后面
=∫ (0-->1) 1/(x+1/x)^3 d(x+1/x)
=(-1/2)(x+1/x)^(-2) [0-->1]
=-1/8