[SOLVED] question on expectations and cumulative distribution functions

I want to prove that $\displaystyle E(X) = \int_{0}^{\infty} (1-F_X(X))\, dx $ is true with the only information given being that X is a continuous RV such that P(X > 0) = 1.

The hint that I've received from my teacher is to replace $\displaystyle F_X(X) $ with the integral of the pdf, to get $\displaystyle E(X) = \int_{0}^{\infty} (1- \int_{0}^{\infty} f_T(t)\, dt )\, dx $ (using dummy variable t) and then switch the order I do the integrals.

Also, with $\displaystyle E(X) = \int_{-\infty}^{\infty} xf(x)\, dx $ (the definition of E(X)) I'm not sure how I would integrate this without knowing exactly what f(x) is.

Thanks in advance for any help :)