probability - How do I show that $E(g(X))= int_{-infty}^infty g(x)f(x),dx$?

Given that $X$ is a continuous random variable with pdf $f$, and $g(x)$ is a nonnegative function.

How do I show that
$$E(g(X))= \int_{-\infty}^\infty g(x)f(x)\,dx$$
using the fact that $E(X) = \int_0^\infty P(X>x)\, dx$.

I attempted to prove this by plugging in g(X) into the second equation instead of just X. And then I took the inverse of g to come up with just a cdf of X, then I rewrote the cdf to its equivalent integral form, giving me an expression with double integral. I have no idea how to move on from here.

Can anyone help me?