Question

一道求不定积分的题：x^3 / （x^2+1） dx = 应该是用换元法计算.

Answer 1

设t=x^2+1
∫x^3/(x^2+1) dx
= ∫(x^2)/2(x^2+1) d(x^2+1)
= ∫(t-1)/(2t) dt
= ∫(1/2)dt - ∫(1/2t)dt
= t/2 - (1/2)·lnt + C .
= (x^2+1)/2 - （1/2)·ln(x^2+1) + C.