another integration problem

It can be solved by using two times by parts

ſ1 x^2 e^x dx
0
=ſ1 x^2 d(e^x)
0
=[(1)^2 e^(1)] - [(0)^2 e^(0)] - ſ1 2x e^x dx
0
=e - 2ſ1 x e^x dx
0
=e - 2 {ſ1 x d(e^x)}
0
=e - 2 {[(1) e^(1)] - [(0) e^(0)] - ſ1 e^x dx}
0
=e - 2 {e - [e^1 - e^0]}
=e - 2 (1)
= e - 2

Use integration by parts

ſ x^2 e^x dx
= ſ x^2 d(e^x)

[since d(e^x)/dx = e^x, so d(e^x) = e^x dx ]

= x^2 (e^x) - ſ e^x d(x^2)
= x^2 (e^x) - ſ e^x * 2x dx
= x^2 (e^x) - ſ 2x d(e^x)
= x^2 (e^x) - [ 2x * (e^x) - ſ e^x d(2x) ]
= x^2 (e^x) - 2x * (e^x) + ſ 2e^x dx
= x^2 (e^x) - 2x * (e^x) + 2e^x +C

so,
ſ1 x^2 e^x dx
0

= [ x^2 (e^x) - 2x * (e^x) + 2e^x ] 1
0
= e - 2e +2e -0 -0 +2
= e + 2

2007-10-15 00:34:47 補充：
sorry, should bee - 2e +2e -0 +0 -2= e - 2