I have $F(x)$, a non-negative function, and a definite integral: $$\int_{0}^{\infty}F(-x)dx$$
How do I use substitution of variable?
If I do this with $u = -x$, then $dx = -du$ and with new limits of integration: $0$ and $-\infty$ resulting in: $$-\int_{-\infty}^{0}F(u)du$$ Now, I have a wrong answer because $F$ is non-negative, but the integral will be negative.
What's missing here?
Your new integration limits over $u$ are reversed: $$\int_0^\infty F(-x)\,\text{d}x =-\int_0^{-\infty} F(u)\,\text{d}u =\int_{-\infty}^0 F(u)\,\text{d}u.$$