Question

不定积分简单问题求解

不定积分简单问题求解横线部分的ABC该如何求？我写的对吗？求解

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Answer 1

let
(x^3+4x^2+x)/[(x+2)^2.(x^2+x+1)]≡A/(x+2)+B/(x+2)^2 +(Cx+D)/(x^2+x+1)
=>
x^3+4x^2+x≡A(x+2)(x^2+x+1)+B(x^2+x+1) +(Cx+D)(x+2)^2
x=-2:
-8+16-2= B(4-2+1)
3B=6
B=2
coef. of constant:
2A+B+4D =0
A+2D =-1 (1)
x^3+4x^2+x≡A(x+2)(x^2+x+1)+B(x^2+x+1) +(Cx+D)(x+2)^2
x=1:
1+4+1 =A(1+2)(1+1+1) +B(1+1+1)+(C+D)(1+2)^2
6=9A+3B+9C+9D
9A+9C+9D=0
A+C+D =0 (2)
x=-1:
-1+4-1≡A(-1+2)(1-1+1)+B(1-1+1) +(-C+D)(-1+2)^2
2=A+B-C+D
A-C+D=0 (3)
(2)-(3)
=> C=0
(1)-(2)
D=-1
from (1)
A+2D =-1
A-2=-1
A=1
ie
(x^3+4x^2+x)/[(x+2)^2.(x^2+x+1)]≡ 1/(x+2)+2/(x+2)^2 -1/(x^2+x+1)