TITLE: $\int_{-\pi/2}^{\pi/2} \frac{\sin^{2012}{x}}{\left(1+ \alpha^x\right)\left(\sin^{2012} {x}+\cos^{2012}{x}\right)}\;{dx} $
QUESTION [10 upvotes]: For $\alpha\in\mathbb{R^+}$, evaluate
$$\displaystyle \int_{-\pi/2}^{\pi/2} \frac{\sin^{2012}{x}}{\left(1+ \alpha^x\right)\left(\sin^{2012}  {x}+\cos^{2012}{x}\right)}\;{dx} $$
Can I have a hint on this? I tried substituting $x=-t$ but got nowhere.

REPLY [2 votes]: This can be solved simply by using reflection. In fact the substitution you made namely $x\to -t$ shall lead you to the solution. You simply need to use the reflection property twice.
$$I=\int_{-\pi/2}^{\pi/2}\dfrac{1}{1+\alpha^x}\underbrace{\biggl(\dfrac{\sin^{2012}x}{\sin^{2012}x+\cos^{2012}x}\biggr)}_{\text{even function}}\mathrm dx = \int_{-\pi/2}^{\pi/2}\dfrac{\alpha^x}{1+\alpha^x}\biggl(\dfrac{\sin^{2012}x}{\sin^{2012}x+\cos^{2012}x}\biggr)\mathrm dx\\\implies 2I=\int_{-\pi/2}^{\pi/2}\dfrac{\mathrm dx}{1+\tan^{2012}x}=2\int_{0}^{\pi/2}\dfrac{\mathrm dx}{1+\tan^{2012}x}\implies I=\dfrac{\pi}{4}$$