Question

Help me integrate by parts

Answer 1

\[\int\limits_{0}^{\infty} (e^{-t}t^{n})dt\]

Answer 2

Apparently the answer is n! if that helps, idk how to get to that

Answer 3

You have to get a recurrence formula, or do integration by part n times.

Answer 4

What's this? The answer is a gamma function

Answer 5

I have to do it via integration by parts, and I have no idea how to succeed

Answer 6

Would I let u=t^n, or u=e^-t

Answer 7

Yes

Answer 8

wow, which one???

Answer 9

If you call
\[
I_n=\int_0^\infty t^n e^{-t}dt
\]

Answer 10

When you do the first integration by part, you get
\[
I_n=n I_{n-1}
\]

Answer 11

You can see where the n! would come from

Answer 12

oh!

Answer 13

wait

Answer 14

\[u=t^{n}\]
\[\frac{ du }{ dt } = nt^{n-1}\]
\[\frac{ dv }{ dt }= e^{-t}\]
\[v=-e^{-t}\]

\[[-e^{-t}t^{n}] + n \int\limits_{}^{} e^{-t}t^{n-1}\]

the left hand side will always become 0, meaning you will end up with n(n-1)(n-2)...etc

thanks!

Answer 15

Yes, That is it