Solved: Integration of x^(2n+1)e^(-x^2) (nεN)

[Saitama]

Homework Statement

\int_{0}^{∞} x^{2n+1}.e^{-x^2}dx
is equal to (n ε N)
a)n!
b)2(n!)
c)n!/2
d)(n+1)!/2

Homework Equations

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]

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

 

[Saitama]

Hello uart!

Hi Pranav. Make the substitution u = x^2 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:
\frac{1}{2} \int_{0}^{∞} u^n \cdot e^{-u}du
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]

Hello uart!I have already tried that substitution and i end up with this:
\frac{1}{2} \int_{0}^{∞} u^n \cdot e^{-u}du
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]

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]

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!