Okay, this is the final part in the above series of posts on some identities related to definite integrals (before I get too lazy and forget to post the same).

So, what is the magic identity? Here it is.

\displaystyle (3) \int_{-a}^{a} f(x) \, dx = \int_0^a \left( f(x) + f(-x)\right) \, dx

Proof: Let t = -x in the second integral on the right hand side. Then, we have \displaystyle \int_0^a f(-x) \, dx = - \int_0^{-a} f(t) \, dt = \int_{-a}^0 f(x) \, dx, and combining this with the first integral on the right hand side yields the desired result.

Now, apply the above identity to the “difficult” integral in problem (6) from the Integration Bee, Challenging Integrals post to evaluate the integral. The solution turns out to be an easy one. The answer is \pi /4, just in case you need to verify.