Here's my stab at it. Denote the integral by . Because of the symmetry this integral is

Under the change of variable , the integral becomes

.

Now replace the dummy variable with , so

.

Now add giving

Now introduce your hint (sort of)

so

.

Note that gives and gives so with that and noting that

gives .

Thus, (3) becomes

and solving for gives

.

Whew! I think I need to lie down.