Question

Convergent/Divergent integral---
Integrate from [negative infinity, 0] (x^2)*(e^x)
I'll actually try to quickly post below how far I've gotten, to see if I'm right so far....

Answer 1

\[\lim_{t \rightarrow -\infty} \int\limits_{t}^{0} x^2e^xdx\]

Answer 2

\[\lim_{t \rightarrow -\infty} -1/2 - t^2e^t + 1/2(e^t)\]

Answer 3

I don't think you integrated right

Answer 4

I integrated by parts...

Answer 5

which gave me =
\[x^2e^x - (1/2)\int\limits_{t}^{0}e^xdx\]
Everything evaluated at t and 0...

Answer 6

\[\int x^2e^xdx\]\[u=x^2\implies du=2xdx\]\[dv=e^xdx\implies v=e^x\]\[\int x^2e^xdx=x^2e^x-2\int xe^xdx\]

Answer 7

ahhh.... maybe I did integrate wrong....hmm. Messed up on the 2xdx part...

Answer 8

yeah you did some non-existent u-substitution thing in your head I guess..

Answer 9

so what should the integral be?

Answer 10

Mkay... so I'm left with
\[\lim_{t \rightarrow -\infty} 2 - t^2e^t + 2e^t t - 2e^t\]

Answer 11

Which leaves me to .... have to use the LH rule? I'm thinking...

Answer 12

yeah, but you gotta do a little substitution befor you can use it

Answer 13

\[\lim_{t \rightarrow -\infty} 2 - t^2e^t + 2e^t t - 2e^t\]\[=2-\lim_{t \rightarrow -\infty} t^2e^t + 2e^t t - 2e^t\]let \(n=-t\) and what do you get?