一道微积分的题

Question

求不定积分:(1+2x^2)/x^2(1+x^2)dx

Answer 1

 
泪笑为您解答，
如若满意,请点击[采纳为满意回答];如若您有不满意之处,请指出,我一定改正!
希望还您一个正确答复!
祝您学业进步!

Answer 2

∫ (1+2x^2)/[x^2(1+x^2)] dx
= 2∫(1+x^2)/[x^2(1+x^2)] dx - ∫ dx/[x^2(1+x^2)]
= 2∫ dx/x^2 -∫ dx/[x^2(1+x^2)]
= -2/x -∫ dx/[x^2(1+x^2)]
= -2/x -∫ dx/x^2 + ∫ dx/(1+x^2)
= -2/x +1/x + arctanx + C
= -1/x + arctanx + C

let
1/[x^2(1+x^2)] ≡ A1/x +A2/x^2 + (B1x+B2)/(1+x^2)
1≡ A1x(1+x^2) +A2(1+x^2) + (B1x+B2)x^2
put x=0, A2=1
coef. of x^2
A2+B2 =0
B2=-1
coef. of x
A1=0
coef. of x^3
A1+B1=0
B1=0

1/[x^2(1+x^2)] ≡ 1/x^2 -1/(1+x^2)本回答被提问者采纳