I want to prove the following :
$$E\lvert X \rvert^{\alpha}=\int_{0}^{\infty} P\left\{\lvert X \rvert^{\alpha}>x\right\} dx=\alpha \int_{0}^{\infty} x^{\alpha-1} P\left\{\lvert X \rvert>x\right\} dx$$
The first equality is easy to prove. I'm having trouble with the second one. It appears to be some differentiation technique has to be used but I don't have a clue on how to proceed. Any help would be much appreciated.
Hint: Try a change of variables. Say $u=x^{1/\alpha}$. Then $dx = \alpha u^{\alpha-1}du$.