For any real-valued random variable $X$ with cdf $F$ [it is well-known][1] that $F^{-1}(U)$ has the same law than $X$ when $U$ is uniform on $(0,1)$. Therefore $$E(X)=E(F^{-1}(U))=\int_0^1 F^{-1}(u)\mathrm{d}u,$$
as long as $E(X)$ exists. This is true for a general $F$, taking $F^{-1}$ to be the left-continuous inverse of $F$ when it is not invertible.

  [1]: http://en.wikipedia.org/wiki/Inverse_transform_sampling