Homework Help: Help for integration

Saitama · Sep 4, 2012

1. The problem statement, all variables and given/known data
[tex]\int_{0}^{∞} x^{2n+1}.e^{-x^2}dx[/tex]
is equal to (n ε N)
a)n!
b)2(n!)
c)n!/2
d)(n+1)!/2

2. Relevant equations

3. The attempt at a solution
I can go on solving this by using n=1 or n=2. But i want to do it by a correct method. Is there a proper way to do it? I am having no idea, how should i begin?

uart · Sep 4, 2012

Hi Pranav. Make the substitution [itex]u = x^2[/itex] and see if you can manipulate it in terms of the gamma function.

Saitama · Sep 4, 2012

Hello uart!

uart said: ↑

Hi Pranav. Make the substitution [itex]u = x^2[/itex] and see if you can manipulate it in terms of the gamma function.

I have already tried that substitution and i end up with this:
[tex]\frac{1}{2} \int_{0}^{∞} u^n \cdot e^{-u}du[/tex]
Gamma function? I guess i have never heard of it.
Any other way to solve it? If there's no way, i would try to see what this "gamma function" is.

uart · Sep 4, 2012

Pranav-Arora said: ↑

Hello uart!

I have already tried that substitution and i end up with this:
[tex]\frac{1}{2} \int_{0}^{∞} u^n \cdot e^{-u}du[/tex]
Gamma function? I guess i have never heard of it.
Any other way to solve it? If there's no way, i would try to see what this "gamma function" is.

Yes, that result is correct. That's good as that makes it equal 1/2 Gamma(n+1). Look up the gamma function and its relation to factorial and you'll find that you basically have the answer there. http://en.wikipedia.org/wiki/Gamma_function

A_B · Sep 4, 2012

I used a different method ;integrating x*exp(-ax^2), then differentiation wr to a several times. You'll quickely find the pattern. Then you can prove by induction. I get a result that's not in your options. (but it's late, so I could be wrong)

A_B

Saitama · Sep 4, 2012

uart said: ↑

Yes, that result is correct. That's good as that makes it equal 1/2 Gamma(n+1). Look up the gamma function and its relation to factorial and you'll find that you basically have the answer there. http://en.wikipedia.org/wiki/Gamma_function

Thanks a lot uart!