Question

x^3+1/(x^4+2x^2+1)的不定积分

Answer 1

详细解答过程如下图片：

Answer 2

x^3+1
=x(x^2 +1) -x +1
∫ (x^3+1)/(x^4+2x^2+1) dx
=∫ (x^3+1)/(x^2+1)^2 dx
=∫ x/(x^2+1) dx - ∫ (x-1)/(x^2+1)^2 dx
=∫ x/(x^2+1) dx - (1/2)∫ 2x/(x^2+1)^2 dx + ∫ dx/(x^2+1)^2
= (1/2)ln|x^2+1| + (1/2)[ 1/(x^2+1)] +∫ dx/(x^2+1)^2
= (1/2)ln|x^2+1| + (1/2)[ 1/(x^2+1)]+(1/2)[ arctanu +x/(1+x^2)] + C
consider

let
x=tanu
dx=(secu)^2 du
∫ dx/(x^2+1)^2
= ∫ (secu)^2 du/(secu)^4
= ∫ (cosu)^2 du
=(1/2)∫ (1+cos2u) du
=(1/2)[ u +(1/2)sin2u] + C'
=(1/2)[ arctanu +x/(1+x^2)] + C'