Originally Posted by

**Krizalid** The integral is easy, it's an obvious arctangent. Now, why am I sayin' this, 'cause you can "see" it. First, take $\displaystyle \arctan \left( \frac{\sin x}{2} \right)$ and contemplate its derivative, which is $\displaystyle \frac{\dfrac{\cos x}{2}}{1+\dfrac{\sin ^{2}x}{4}}=\frac{2\cos x}{4+\sin ^{2}x},$ and the conclusion follows quickly.

-----

I suggest you to find nice integrals, like hard ones. (Of course, when sayin' "hard" I mean "it looks hard but with a simple trick we can kill it.")