# Help for integration

## 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

## 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?

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

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

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!