algebraic derivation involving integration , expectation, and max function

Question

please help me prove this:

$K,R_0$ and $L_0$ are constants, $R$ is a random variable , + superscript indicates $max(.,0)$ and $R_0$=E(R)

note: this comes from me trying to understand the maths of getting from equation 3.2 to 3.4a in the following paper: www.javaquant.net/papers/ConvexityConundrum.pdf

Answer 1

Hint: $R-x$ is positive if $x < R$. If $x \geq R$, then $(R-x)^+$ is equal to 0.

Therefore the upper bound for the integral is $R$:

$\ldots+2\int_K^R L_0E\left( R-x\right) \ dx$