ISI2014-DCG-12

Question

The integral $$\int _0^{\frac{\pi}{2}} \frac{\sin^{50} x}{\sin^{50}x +\cos^{50}x} dx$$ equals

• A. $\frac{3 \pi}{4}$
• B. $\frac{\pi}{3}$
• C. $\frac{\pi}{4}$
• D. none of these

Answer 1

Let $\displaystyle I=\int_{0}^{\frac{\pi}{2}} \frac{\sin^{50}x}{\sin^{50}x+\cos^{50}x}dx\qquad \to (1)$

By using property of definite integration

$\displaystyle \int_{0}^{a}f(x)dx=\int_{0}^{a}f(a-x)dx$

$\displaystyle I=\int_{0}^{\frac{\pi}{2}} \frac{\cos^{50}x}{\cos^{50}x+\sin^{50}x}dx\qquad \to (2)$

Adding $(1)$ and $(2),$

$\displaystyle 2I=\int_{0}^{\frac{\pi}{2}} \frac{\sin^{50}x+\cos^{50}x}{\sin^{50}x+\cos^{50}x}dx$

$\displaystyle \implies 2I=\int_{0}^{\frac{\pi}{2}}1\;dx$

$\displaystyle \implies 2I=\left[x\right]_{0}^{\frac{\pi}{2}}$

$\implies 2I=\left[ \frac{\pi}{2}\right]$

$\implies I=\frac{\pi}{4}$

So, answer is (C).