How to integrate e to the power to the power?

How should I integrate?

$\int_0^\infty e^{-x^{1/3}}dx$

I think this is a simple question for the experts. But a bit hard to tell Google what I want. So, thanks for your help! :)

And this looks terrifying to me.

Hint: Try the change of variable $u=x^{\frac{1}{3}}$ (so $dx=3u^2du$).
• What do you mean by "U- substitution"? The reason for doing a change of variables like this is simply "the bit in the exponent looks tricky, let's simplify it by defining it to just be something new, like $u$". – Sharkos May 18 '13 at 0:44