用分部积分法求∫(e~e^2)xln^2xdx
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解决时间 2021-03-24 11:34
- 提问者网友：战皆罪
- 2021-03-23 17:49
用分部积分法求∫(e~e^2)xln^2xdx
最佳答案
- 五星知识达人网友：北城痞子
- 2021-03-23 18:32
∫xln^2xdx
=1/2 *∫ln^2xdx^2
=1/2 *ln^2x *x^2 -∫1/2 *x^2 dln^2x
=1/2 *ln^2x *x^2 -∫1/2 *x *2lnx dx
=1/2 *ln^2x *x^2 -1/2 ∫lnx d(x^2)
=1/2 *ln^2x *x^2 -1/2 * lnx *x^2 +∫1/2 x^2 d(lnx)
=1/2 *ln^2x *x^2 -1/2 * lnx *x^2 +∫1/2 x dx
=1/2 *ln^2x *x^2 -1/2 * lnx *x^2 +1/4 x^2
代入上下限e^2和e
=2e^4 -e^4+1/4 e^4 -1/2e^2 +1/2e^2 -1/4e^2
=5/4 e^4 -1/4e^2
=1/2 *∫ln^2xdx^2
=1/2 *ln^2x *x^2 -∫1/2 *x^2 dln^2x
=1/2 *ln^2x *x^2 -∫1/2 *x *2lnx dx
=1/2 *ln^2x *x^2 -1/2 ∫lnx d(x^2)
=1/2 *ln^2x *x^2 -1/2 * lnx *x^2 +∫1/2 x^2 d(lnx)
=1/2 *ln^2x *x^2 -1/2 * lnx *x^2 +∫1/2 x dx
=1/2 *ln^2x *x^2 -1/2 * lnx *x^2 +1/4 x^2
代入上下限e^2和e
=2e^4 -e^4+1/4 e^4 -1/2e^2 +1/2e^2 -1/4e^2
=5/4 e^4 -1/4e^2
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