Integration change of variables

Question

I have the following integral to solve $$\int_a^b e^{\sigma z}e^{-\frac{1}{2}z^2}dz $$ where $\sigma,a,b$ are some constants. I rewrite this as $$e^{\sigma^2/2}\int_a^b e^{-\frac{1}{2}(z-\sigma)^2}dz $$ By substituting $u=z-\sigma$, we have $du=dz$ and the limits become $a+\sigma$ to $b+\sigma$. Then, $$e^{\sigma^2/2}\int_{a+\sigma}^{b+\sigma}e^{-\frac{1}{2}u^2}dz = e^{\sigma^2/2}\big(N(b+\sigma)-N(a+\sigma) \big) $$ where $N(\cdot)$ is the normal density CDF. Am I using the correct limits (or is it $a-\sigma, b-\sigma$ for the lower and upper limits respectively)?

Answer 1

In your last integral, you forgot to change $dz$ to $du$. Maybe that is why you are getting confused. When $z$ is $a$, $u$ is $a-\sigma$.

When $z$ is $b$, $u$ is $b-\sigma$.