The value of $\frac{8}{\pi} \int_{0}^{\frac{\pi}{2}} \frac{(\cos x)^{2023}}{(\sin x)^{2023}+(\cos x)^{2023}} d x$ is

JEE Main · 2023 · 24 Jan 2023 · Shift 2 · Definite Integrals

The value of

\frac{8}{π} \int_{0}^{\frac{π}{2}} \frac{( c o s x ) ^{2023}}{( s i n x ) ^{2023} + ( c o s x ) ^{2023}} d x

Answer: 2

Let

I = \frac{8}{π} \int_{0}^{π /2} \frac{cos ^{2023} x}{sin ^{2023} x + cos ^{2023} x} d x

. Replacing

x

\frac{π}{2} - x

turns the integrand into

\frac{sin ^{2023} x}{sin ^{2023} x + cos ^{2023} x}

; adding the two forms,

2 I = \frac{8}{π} \int_{0}^{π /2} 1 d x = \frac{8}{π} \cdot \frac{π}{2} = 4

. Therefore

I = 2.

JEE Main · 2023 · 24 Jan 2023 · Shift 2 · Integer answer

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