Consider the Fredholm integral equation:
$f(x)=\int_{0}^{\infty}e^{-xg(y)}dy$. I want to solve it in $L^2$, solving for the unknown $g(x)$. I tried expanding the function as a Taylor Series. This yields $exp(-xg(y))=\sum_{n=0}^{\infty}\frac{(-1)^n (x g(y))^n}{n!}$, so we can integrate $f(x)=\int_{0}^{\infty}\sum_{n=0}^{\infty}\frac{(-1)^n (x g(y))^n}{n!}$. Now can we just integrate term by term? I am unsure how to go from here to the solution.