I don't know what you consider easy and what you know, but you can skirt the complex analysis by using the Laplace transform.

Let

And note that f is an EVEN function! This will be important later.

If we take the Laplace transform we get

Now if we use partial fraction decomposition with respect to x we get that

So if we take the integral we get

So if we take the Inverse Laplace transform we get

But since we know f is even we get