This is from Sheldon Ross' text, "A First Course in Probability":
Use the following result that, for a nonnegative random variable Y,
$E[Y] = \displaystyle\int\limits_{0}^{\infty}P(Y > t)dt$
to show that, for a random variable X,
$E[|X|^{n}] = \displaystyle\int\limits_{0}^{\infty}nx^{n-1}P(|X|\geq x)dx$
Do I have to do an integration by parts somewhere?
On
No need for integration by parts. As Robert Israel wrote, just change of variable such as the following: $E[X^n] = \displaystyle \int_0^\infty P\{X^n>t\}\, dt = \displaystyle \int_0^\infty nx^{n-1}P\{X^n>x^n\}\, dx = \displaystyle \int_0^\infty nx^{n-1}P\{X>x\}\, dx$ $\\$ and the rest follows. Notice that the variable change was $t=x^n.$
If you already have the result about $E[Y]$, you just need a change of variables.