Write this integral in terms of gamma function

[eprparadox]

Hey!

So I'm self studying mary boas's mathematical methods book and I've come across this integral:

<br /> \int _{0}^{\infty }e^{-x^4}dx<br />

and I'm suppose to write this using the gamma function. The hint given states to let x^4 = u. And the answer is:

<br /> \Gamma \left( \dfrac {5} {4}\right) <br />

I tried substituting u = x^4 and du = 4x^3dx, but that doesn't give the correct answer.

I'm a bit confused as to how the book got that answer. Any ideas would be great.

Thanks!

 

[Strum]

After substitution you should have something you can express as $t \Gamma(t)$ which you can rewrite using the relation $t\Gamma(t) = \Gamma(t+1)$.