5.4.2 problem 35

Let $g\left(Z\right)=\max (Z-c,0)$. We have $g\left(Z\right)=0$ for $Z<c$ and $g\left(Z\right)=Z-c$ for $Z>c$.

Then

since the expression inside the integral is 0 for $k<c$. Next we have

with the first equality following from splitting the integral via subtraction and expanding out $\phi$ for the left integral, the second equality comes from observing that the antiderivative of $\mathrm{k\phi }\left(k\right)$ is $-\phi \left(k\right)$ and that we can change the limits of the right integral due to the symmetry of tail areas of the curve of $\phi$, and the last equality comes from applying the definitions of the PDF and CDF of the standard normal distribution.