Integration by parts with continuous random variables

Question

Show that for any continuous positive random variable $X$ with $F(x) = F_X(x)$

we have $$EX =\int_{0}^{\infty} (1 − F(x)) \text{ d}x\text{.}$$

[Hint: use integration by parts on $(1 − F(x)) · 1$.]

I have no idea how to use the hint, could someone help for this? Thanks

Answer 1

Hint:

$$\int_{(0,b]}x^aF(x)dx=-b^a(1-F(b))+a\int_{(0,b]}x^{a-1}(1-F(x))dx$$

If you need more hints feel free to ask.