1. Evaluate Integral

Evaluate Integral

$\displaystyle \int_0^\frac{\pi}{2}\frac{sin^{25} x}{\cos^{25} x+\sin^{25} x}dx$

2. Originally Posted by perash
Evaluate Integral

$\displaystyle \int_0^\frac{\pi}{2}\frac{sin^{25} x}{\cos^{25} x+\sin^{25} x}dx$
More generally:

$\displaystyle \int_0^{\pi /2} {\frac{{\sin ^n x}} {{\cos ^n x + \sin ^n x}}\,dx} = \frac{\pi } {4}.$ (No matter what $\displaystyle n$ is.)

Let $\displaystyle \varphi =\int_0^{\pi /2} {\frac{{\sin ^n x}} {{\cos ^n x + \sin ^n x}}\,dx}.$ Substitute $\displaystyle u = \frac{\pi } {2} - x,$

$\displaystyle \varphi = \int_0^{\pi /2} {\frac{{\sin ^n x}} {{\cos ^n x + \sin ^n x}}\,dx} = \int_0^{\pi /2} {\frac{{\cos ^n x}} {{\cos ^n x + \sin ^n x}}\,dx} .$

This yields $\displaystyle 2\varphi = \frac{\pi } {2}$ and the rest follows.