How can I evaluate this integral using the delta function?

Question

I am unsure how to answer this question using the delta function. I am unfamiliar with the summation inside of the integral

Answer 1

The sum of delta functions is sometimes called a Dirac comb. Here, this sum is $1$ when $x$ is a nonnegative even integer ($0,2,4,\dots$) and $0$ otherwise. The integral is therefore turned into an infinite sum, since $e^{-x}$ is only evaluated where the comb returns $1$: $$\int_{-\infty}^\infty e^{-x}\sum_{n\ge0}\delta(x-2n)=\sum_{n=0}^\infty e^{-2n}=\frac1{1-e^{-2}}$$