I'm trying to solve $$\int_{-\infty}^\infty\frac{1}{\sqrt{2\pi}\sigma}e^{-\left(\frac{(x - \mu)^2}{2\sigma^2} + kx\right)}dx$$
This is very similar to the integral of the normal PDF and it looks fairly similar to a Gauss integral but I'm not sure what substitutions to make in order to get there.
Hint: complete the square in the exponent, and then with a linear substitution, you will reduce the problem to the integral of the PDF of the normal distribution.