I am trying to calculate this:
Let $X \sim \operatorname{Gamma}(\lambda, k)$. If $0 < a < \lambda$, find $E(e^{aX})$.
I have attempted this by integration, I inserted a picture of my integral:
Because of the form of the integral I believe that I should take a constant out to change the $\lambda$ to $a-\lambda$ as that is what appears in the exponential, and then get the integral of the PDF of a $\operatorname{Gamma}(a-\lambda,k)$ random variable which integrates to $1$ and simplifies the expression, but I am not sure how to do this.