Substituting limit in integral bound and integrand random variable

Question

Let $X$ be a continuous random variable taking non-negative values only. To prove: $\mathbb{E}(X)=\int_0^\infty (1-F_X(x)) dx$ where $F_X$ is the distribution function of $X$. I have derived the following expression: $\mathbb{E}(X)=\lim_{t\to\infty} \int_0^t (1-\frac{1}{F_X(t)} F_X(x))dx$. My question is: can I replace the upper bound with $\infty$ and substitute $F_X(t)=1$, als $\lim_{t\to\infty} F_X(t)=1$? If so, why does this work?