Integration Problem: Solving $\int_{0}^{\infty}x^3.e^{-x^2}\mathrm{d}x$

[qspeechc]

Homework Statement

$$\int_{0}^{\infty}x^3.e^{-x^2} \mathrm{d}x$$

The Attempt at a Solution

I have tried substitution u=x^2, u=x^3; integration by parts; squeeze theorem; partial fraction decomp; taylor series expansion- but nothing seems to work. I know the limit of $$x^3.e^{-x^2}$$ as x tends to infinity is zero, but that doesn't help.
Any help please?

 

[neutrino]

I have tried substitution u=x^2

That should work.

 

[qspeechc]

Well, ok.

Let u = x^2 , then: du = 2x.dx. And then what? We have an x^3 in the integration, so I don't see how it works.

 

[neutrino]

But x³ = x.x², isn't it?

 

[qspeechc]

Yes, but then we would have:

$$x^2.e^{-u}.du$$ or am I missing something?

 

[neutrino]

$$x^2.e^{-u}.du$$ or am I missing something?

You just did the sub u=x^2 a couple of steps ago.

 

[qspeechc]

AHA! Oh thank-you neutrino! I can't believe I never saw that! Gee, I feel like an idiot! Thank-you again!

 

[neutrino]

AHA! Oh thank-you neutrino! I can't believe I never saw that! Gee, I feel like an idiot! Thank-you again!

You're welcome. Make a hobby out of solving integrals (if you're into those kind of things), and you'll start recognising the methods with just a look at the integral. (For some of them, at least. )

 

[qspeechc]

Actually I quite dislike calculus. I find it dry and boring. Or atleast that's the way my first year course presents it. But I guess you are right, I need to do more calculus problems if I want to be a mathematician , which I do.