x∫─────dx＝怎么解答啊1－x4

Question

x ∫─────dx＝_____________怎么解答啊 1－x4

Answer 1

令u = x²,du = 2x dx
∫ x/(1 - x^4) dx
= ∫ x/(1 - u²) * du/(2x)
= (1/2)∫ du/[(1 + u)(1 - u)]
= (1/2)∫ [(1 - u) + (1 + u)]/[2(1 + u)(1 - u)] du
= (1/4)∫ [1/(1 + u) + 1/(1 - u)] du
= (1/4)∫ d(1 + u)/(1 + u) - (1/4)∫ d(1 - u)/(1 - u)
= (1/4)ln|1 + u| - (1/4)ln|1 - u| + C
= (1/4)ln|(1 + u)/(1 - u)| + C
= (1/4)ln|(1 + x²)/(1 - x²)| + C