Probability $P(A < B)$ 
Given two independent and continuous random variables $A$ and $B$ with cumulative distributions $F_A$ and $F_B$, show that $$P(A<B) = \int_{-\infty}^{\infty} F_A(x)\, F'_B(x)\,dx.$$ 

Is this something obvious and available in text books ?

Let the density functions be $f_A(x)$ and $f_B(y)$. So the joint density is $f_A(x)f_B(y)$.
To find the probability that $A\lt B$, we integrate the joint density over the part of the plane made up of points $(x,y)$ such that $x\lt y$. 
So we are integrating over the part of the plane above the line $x=y$. 
This double integral can be expressed as the iterated integral
$$\int_{y=-\infty}^\infty\left( \int_{x=-\infty}^y f_A(x)\,dx\right) f_B(y)\,dy.$$
The innner integral is $F_A(y)$, and $f_B(y)=F'_B(y)$, so we end up with
$$\int_{-\infty}^\infty F_A(y)F_B'(y)\,dy.$$
What you quoted uses a different name for the dummy variable of integration. 
