Question

find the integral of xe^x from -infinity to 0

Answer 1

\[\int\limits_{-\infty}^{0}xe ^{x}dx\]

Answer 2

\[ \large \int_{-\infty}^0xe^x\,dx=\lim_{b\to+\infty}\int_{-b}^0xe^x\,dx \]

Answer 3

use integration by parts

Answer 4

I used integration by parts and got stuck at a particular place towards the end.

Answer 5

u = x,u'=1,v=e^x,v'=e^x

Answer 6

\[ \large \begin{alignat*}{2}
u&=x &\qquad dv=e^x\\ du&=dx&\qquad v=e^x
\end{alignat*} \]
so
\[ \large \int xe^x\,dx=xe^x-\int e^x\,dx=xe^x-e^x \]

Answer 7

ok so then would you take the limit of xe^x - e^x?

Answer 8

right
\[ \large \lim_{b\to+\infty}\int_{-b}^0xe^x\,dx=
\lim_{b\to+\infty}\left[ xe^x-e^x\right]_b^0 \]

Answer 9

The main thing I don't understand is when you are solving with integration by parts is why you son't leave the domain for the integrals from -infinity to 0

Answer 10

just to save time when typing

Answer 11

because I would think you then would have to evaluate e^x from -infinity to 0 before you could take the limit. of xe^x - e^x

Answer 12

ok but uv' evaluated at -inf to 0 = xe^x - [e^x evaluated at -inf to 0] ?

Answer 13

\[ \large \int_{-\infty}^0xe^x\,dx=xe^x|_{-\infty}^0-\int_{-\infty}^0e^x\,dx \]
i don't see where is the problem.

Answer 14

\[ \large =xe^x|_{-\infty}^0-e^x|_{-\infty}^0=[xe^x-e^x]_{-\infty}^0 \]

Answer 15

ah I think I understand now. thinking into it more. I appreciate your help.

Answer 16

u r welcome