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.