Question

The value of ${\int }_0^{\frac{\pi }{2}}\frac{{\cos }^{\frac{5}{3}}x}{{\cos }^{\frac{5}{3}}x+{\sin }^{\frac{5}{3}}x}dx$ is

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

Answer 1

Answer: (B)

$\frac{\pi }{4}$

Consider, $I={\int }_0^{\frac{\pi }{2}}\frac{{\cos }^{\frac{5}{3}}x}{{\cos }^{\frac{5}{3}}x+{\sin }^{\frac{5}{3}}x}.dx$...(i)

Now, ${\int }_0^{a}f\left(x\right).dx={\int }_0^{a}f(a-x).dx$

Therefore, $I={\int }_0^{\frac{\pi }{2}}\frac{{\cos }^{\frac{5}{3}}(\frac{\pi }{2}-x)}{{\cos }^{\frac{5}{3}}(\frac{\pi }{2}-x)+{\sin }^{\frac{5}{3}}(\frac{\pi }{2}-x)}$

$I={\int }_0^{\frac{\pi }{2}}\left(\frac{{\sin }^{\frac{5}{3}}x}{{\sin }^{\frac{5}{3}}x+{\cos }^{\frac{5}{3}}x}\right)dx$...(ii)

Addition of i and ii gives

$2I={\int }_0^{\frac{\pi }{2}}\left(\frac{{\sin }^{\frac{5}{3}}x}{{\sin }^{\frac{5}{3}}x+{\cos }^{\frac{5}{3}}x}\right)dx$+${\int }_0^{\frac{\pi }{2}}\frac{{\cos }^{\frac{5}{3}}x}{{\cos }^{\frac{5}{3}}x+{\sin }^{\frac{5}{3}}x}.dx$

$2I={\int }_0^{\frac{\pi }{2}}1.dx$

$2I=[x{]}_0^{\frac{\pi }{2}}$

$2I=\frac{\pi }{2}$

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