# Integration Problem

## 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.

I have tried substitution u=x^2
That should work.

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.

But x3 = x.x2, isn't it?

Yes, but then we would have:

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

$$x^2.e^{-u}.du$$ or am I missing something?
You just did the sub u=x^2 a couple of steps ago.

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

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. )

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 :tongue:, which I do.