Question

求此不定积分

Answer 1

(u^3+u^2-u-1)

=(u-1)(u^2+ 2u +1)

=(u-1)(u+1)^2
let
-8u^2/(u^3+u^2-u-1)≡ A/(u-1) +B/(u+1) +C/(u+1)^2
=>
-8u^2 ≡ A(u+1)^2 +B(u-1)(u+1) +C(u-1)
u=1, => A=-2
u=-1, => C = 4
coef. of u^2
A+B = -8
-2+B=-8
B=-6
-8u^2/(u^3+u^2-u-1)
≡-2[1/(u-1)] -6[1/(u+1)] +4[1/(u+1)^2]
∫ 8u^2/(1-u^3-u^2+u) du
=∫ -8u^2/(u^3+u^2-u-1) du
=∫ {-2[1/(u-1)] -6[1/(u+1)] +4[1/(u+1)^2] } du
= -2ln|u-1| -6ln|u+1| + 4arctanu + C