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For any real-valued random variable $X$ with cdf $F$ it is well-known 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.