I want to determine the constant c so that this definite integral evaluates to 1. I have no information about the behaviour of this function in different intervals (e.g. which term inside max() is greater or smaller than the other in different intervals - is not known). |x| is absolute function.
$\mathbb{R}^2$ can be broken up into different regions to make this problem easier. Let's find the first region together.
Let's first assume $x>a$ and $y>b$ (and that $K_1,K_2>0$. Consider $\frac{x-a}{K_1}$ and $\frac{y-b}{K_2}$. When is $\frac{x-a}{K_1}>\frac{y-b}{K_2}$. $$\frac{x-a}{K_1}>\frac{y-b}{K_2} \Rightarrow x-a > \frac{y-b}{K_2}K_1=\frac{y-b}{C}$$ where $C=\frac{K_2}{K_1}$. Then $x>\frac{1}{C}y-\frac{b}{C}+a$. Thus, we've found our first region.
Go about breaking up $\mathbb{R}^2$ into the relevant regions, and this will correspond to breaking up the double integration into a sum of integrations of simple polynomial functions.