∫（x^5+x^4-8）/（x³-x）dx
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解决时间 2021-04-03 07:52
- 提问者网友：低吟詩仙的傷
- 2021-04-03 01:34
∫（x^5+x^4-8）/（x³-x）dx
最佳答案
- 五星知识达人网友：枭雄戏美人
- 2021-04-03 02:37
∫(x^5+x^4-8)/(x^3-x)dx
x^5+x^4-8
=x^2.(x^3-x) + x^4+x^3-8
=x^2.(x^3-x) + x(x^3-x) + x^3+x^2 -8
=x^2.(x^3-x) + x(x^3-x) + (x^3-x) + x^2+x -8
∫(x^5+x^4-8)/(x^3-x)dx
=∫[ x^2 +x +1 + (x^2+x-8)/(x^3-x) ]dx
=(1/3)x^3 + (1/2)x^2 +x +∫ (x^2+x-8)/(x^3-x) ]dx
=(1/3)x^3 + (1/2)x^2 +x +(1/3)∫ (3x^2-1)/(x^3-x) ]dx
+ ∫ (x- 23/3)/(x^3-x) ]dx
=(1/3)x^3 + (1/2)x^2 +x +(1/3)ln|x^3-x| + (1/3)∫ (3x- 23/(x^3-x) ]dx
let
(3x-23)/(x^3-x) ≡A/x + B/(x-1) + C/(x+1)
=>
3x-23 ≡A(x-1)(x+1) + Bx(x+1) + Cx(x-1)
x=0, A=23
x=1, B = -10
x=-1, C= -13
(3x-23)/(x^3-x) ≡ 23/x -10/(x-1) -13/(x+1)
∫(x^5+x^4-8)/(x^3-x)dx
=(1/3)x^3 + (1/2)x^2 +x +(1/3)ln|x^3-x| + (1/3)∫ (3x- 23/(x^3-x) ]dx
=(1/3)x^3 + (1/2)x^2 +x +(1/3)ln|x^3-x|
+(1/3)[ 23ln|x| -10ln|x-1| -13ln|x+1| ) + C
x^5+x^4-8
=x^2.(x^3-x) + x^4+x^3-8
=x^2.(x^3-x) + x(x^3-x) + x^3+x^2 -8
=x^2.(x^3-x) + x(x^3-x) + (x^3-x) + x^2+x -8
∫(x^5+x^4-8)/(x^3-x)dx
=∫[ x^2 +x +1 + (x^2+x-8)/(x^3-x) ]dx
=(1/3)x^3 + (1/2)x^2 +x +∫ (x^2+x-8)/(x^3-x) ]dx
=(1/3)x^3 + (1/2)x^2 +x +(1/3)∫ (3x^2-1)/(x^3-x) ]dx
+ ∫ (x- 23/3)/(x^3-x) ]dx
=(1/3)x^3 + (1/2)x^2 +x +(1/3)ln|x^3-x| + (1/3)∫ (3x- 23/(x^3-x) ]dx
let
(3x-23)/(x^3-x) ≡A/x + B/(x-1) + C/(x+1)
=>
3x-23 ≡A(x-1)(x+1) + Bx(x+1) + Cx(x-1)
x=0, A=23
x=1, B = -10
x=-1, C= -13
(3x-23)/(x^3-x) ≡ 23/x -10/(x-1) -13/(x+1)
∫(x^5+x^4-8)/(x^3-x)dx
=(1/3)x^3 + (1/2)x^2 +x +(1/3)ln|x^3-x| + (1/3)∫ (3x- 23/(x^3-x) ]dx
=(1/3)x^3 + (1/2)x^2 +x +(1/3)ln|x^3-x|
+(1/3)[ 23ln|x| -10ln|x-1| -13ln|x+1| ) + C
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