Question

$\stackrel{\pi /2}{\underset{0}{\int }}\frac{{\sin }^{6}x}{{\cos }^{6}x+{\sin }^{6}x}dx$ is equal to:

• A. $0$
• B. $\pi$
• C. $\frac{\pi }{4}$
• D. $2\pi$

Answer 1

Answer: (C)

$\frac{\pi }{4}$

Let, $I=$$\stackrel{\pi /2}{\underset{0}{\int }}\frac{{\sin }^{6}x}{{\cos }^{6}x+{\sin }^{6}x}dx$.......(1).

or, $I=$$\stackrel{\pi /2}{\underset{0}{\int }}\frac{{\cos }^{6}x}{{\sin }^{6}x+{\cos }^{6}x}dx$ [ Using property of definite integral].......(2).

Now adding (1) and (2) we get,

$2I=\underset{0}{\overset{\frac{\pi }{2}}{\int }}dx$

or, $I=\frac{\pi }{4}$.