Question

高数求解～ (x^2·e^x)/(x+2)^2的不定积分～ 谢谢了。。

Answer 1

简单计算一下即可，答案如图所示

Answer 2

∫(x^2e^x)/(x+2)^2dx
=∫e^x(x^2+4x+4-4x-8+4)/(x+2)^2dx
=∫e^x(x+2)^2-4(x+2)+4)/(x+2)^2dx
=∫e^x[(x+2)^2-4(x+2)+4]/(x+2)^2dx
=∫e^x[1-4/(x+2)+4/(x+2)^2]dx
=∫e^x-4e^x/(x+2)+4e^x/(x+2)^2dx
=e^x-4∫e^x/(x+2)dx+4∫e^x/(x+2)^2dx
=e^x-4∫e^x/(x+2)dx-4∫e^xd[1/(x+2)]
=e^x-4∫e^x/(x+2)dx-4e^x/(x+2)-4∫1/(x+2)de^x
=e^x-4∫e^x/(x+2)dx-4e^x/(x+2)+4∫1/(x+2)de^x
=e^x-4e^x/(x+2)+C

Answer 3

用凑微分法解
∫x^2 * e^x / (x+2)^2
= -∫x^2 * e^x d(x+2)^(-1)
=-x^2 * e^x / (x+2) + ∫e^x * (x^2+2x)/(x+2) dx
上下相消得
=-x^2 * e^x / (x+2) + ∫xe^xdx
继续凑微分
=-x^2 * e^x / (x+2) + ∫xde^x
=-x^2 * e^x / (x+2) + (x-1)e^x + C
合并得
=[(x-2)/(x+2)] * e^x + C