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.

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 the expectation of $X$, whenever it exists, is the same as the expectation of $F^{-1}(U)$: $$E(X)=E(F^{-1}(U))=\int_0^1 F^{-1}(u)\mathrm{d}u.$$ The representation $X \sim F^{-1}(U)$ holds for a general cdf $F$, taking $F^{-1}$ to be the left-continuous inverse of $F$ in the case when $F$ it is not invertible.

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 right-continuous inverse of $F$ when it is not invertible.

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.