Help me integrate by parts

Question

amorfide · Answer

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

amorfide · Answer

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

eliesaab · Answer

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

.sam. · Answer

What's this? The answer is a gamma function

amorfide · Answer

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

amorfide · Answer

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

eliesaab · Answer

Yes

amorfide · Answer

wow, which one???

eliesaab · Answer

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

eliesaab · Answer

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

eliesaab · Answer

You can see where the n! would come from

amorfide · Answer

oh!

amorfide · Answer

wait

amorfide · Answer

$$u=t^{n}$$
$$\frac{ du }{ dt } = nt^{n-1}$$
$$\frac{ dv }{ dt }= e^{-t}$$
$$v=-e^{-t}$$

$$[-e^{-t}t^{n}] + n \int\limits_{}^{} e^{-t}t^{n-1}$$

the left hand side will always become 0, meaning you will end up with n(n-1)(n-2)...etc

thanks!

eliesaab · Answer

Yes, That is it