这几道微积分怎么写？

Question

请给过程

Answer 1

喜欢做，做做更健康。

这个题目或许还要更简单的方法，有空再试试。

供参考，请笑纳。

追答

见笑了。

Answer 2

分部积分公式

本回答被提问者采纳

Answer 3

（12）I = ∫<0, 1>(x+2)de^x = [(x+2)e^x]<0, 1> - ∫<0, 1>e^xdx
= 3e-2 - [e^x]<0, 1> = 2e-1.
（13）I = (1/2)∫<1, e>lnxd(x-1)^2
= (1/2)[(x-1)^2 lnx]<1, e> - (1/2)∫<1, e>[(x-1)^2/x]dx
= (1/2)(e-1)^2 - (1/2)[x^2/2-2x+lnx]<1, e>
= (1/2)(e-1)^2 - (1/2)[e^2/2-2e+5/2] = (e^2-3)/4
（14）I = ∫<0, 2>(x-2)de^x = [(x-2)e^x]<0, 2> - ∫<0, 2>e^xdx
= 2 - [e^x]<0, 2> = 3-e^2.

Answer 4

(12)
∫(0->1)(x+2) e^x dx
=∫(0->1)(x+2) de^x
=[(x+2)e^x ]|(0->1) -∫(0->1) e^x dx
=3e-2 -[e^x]|(0->1)
=3e-2 -(e-1)
=2e-1
(13)
∫(1->e) (x+1) lnx dx
=(1/2)∫(1->e) lnx d(x+1)^2
=(1/2)[(x+1)^2.lnx]|(1->e) -(1/2)∫(1->e) (x+1)^2/x dx
=(1/2)(e+1)^2 -(1/2)∫(1->e) [x+2 + 1/x] dx
=(1/2)(e+1)^2 -(1/2)[(1/2)x^2+2x + ln|x|]|(1->e)
=(1/2)(e+1)^2 -(1/2){ [(1/2)e^2+2e + 1] - [1/2+2+0] }
=(1/2)(e+1)^2 -[ (1/4)e^2+e - 1/4 ]
=(1/2)( e^2 +2e +1) -[ (1/4)e^2+e - 1/4 ]
=(1/4)e^2 +3/4
(14)
∫(0->2)(x-2)e^x dx
=∫(0->2)(x-2) de^x
=[(x-2)e^x]|(0->2) -∫(0->2) e^x dx
=2 -[e^x]|(0->2)
=3-e^2