(x^3-17)/x^2-4x 3x的定积分
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解决时间 2021-02-09 16:51
- 提问者网友：放下
- 2021-02-09 10:01
(x^3-17)/x^2-4x 3x的定积分
最佳答案
- 五星知识达人网友：爱难随人意
- 2021-02-09 11:25
x^3-17
=x(x^2-4x+3) +4x^2-3x-17
=x(x^2-4x+3) +4(x^2-4x+3) +13x -29
∫ (x^3-17)/(x^2-4x+3) dx
=∫ [ x + 4 + (13x-29)/(x^2-4x+3) ]dx
= (1/2)x^2 +4x + ∫ (13x-29)/(x^2-4x+3) dx
= (1/2)x^2 +4x + (13/2)∫(2x-4)/(x^2-4x+3) dx -3∫dx/(x^2-4x+3)
= (1/2)x^2 +4x + (13/2)ln|x^2-4x+3| -3∫dx/[(x-1)(x-3)]
= (1/2)x^2 +4x + (13/2)ln|x^2-4x+3| -(3/2)∫[ 1/(x-1)-1/(x-3) ] dx
= (1/2)x^2 +4x + (13/2)ln|x^2-4x+3| -(3/2)ln|(x-1)/(x-3)| + C
=x(x^2-4x+3) +4x^2-3x-17
=x(x^2-4x+3) +4(x^2-4x+3) +13x -29
∫ (x^3-17)/(x^2-4x+3) dx
=∫ [ x + 4 + (13x-29)/(x^2-4x+3) ]dx
= (1/2)x^2 +4x + ∫ (13x-29)/(x^2-4x+3) dx
= (1/2)x^2 +4x + (13/2)∫(2x-4)/(x^2-4x+3) dx -3∫dx/(x^2-4x+3)
= (1/2)x^2 +4x + (13/2)ln|x^2-4x+3| -3∫dx/[(x-1)(x-3)]
= (1/2)x^2 +4x + (13/2)ln|x^2-4x+3| -(3/2)∫[ 1/(x-1)-1/(x-3) ] dx
= (1/2)x^2 +4x + (13/2)ln|x^2-4x+3| -(3/2)ln|(x-1)/(x-3)| + C
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