Question

The value of $\frac{8}{\pi }\underset{0}{\overset{\frac{\pi }{2}}{\int }}\frac{(\cos x{)}^{2023}}{(\sin x{)}^{2023}+(\cos x{)}^{2023}}dx$ is

Answer 1

Answer: 2

$I=\frac{8}{\pi }\underset{0}{\overset{\frac{\pi }{2}}{\int }}\frac{(\cos x{)}^{2023}}{(\sin x{)}^{2023}+(\cos x{)}^{2023}}dx$......(1)
Using $\underset{0}{\overset{a}{\int }}f(x)dx=\underset{0}{\overset{a}{\int }}f(a-x)dx$
$I=\frac{8}{\pi }\underset{0}{\overset{\frac{\pi }{2}}{\int }}\frac{(\sin x{)}^{2023}}{(\sin x{)}^{2023}+(\cos x{)}^{2023}}dx$........(2)
Adding (1) & (2)
$2I=\frac{8}{\pi }\underset{0}{\overset{\frac{\pi }{2}}{\int }}1dx$
$I=2$