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  4. The value of (8/π) ∫ limits0(π/2) (( cos x)2023/( sin x)2023+( cos x)2023) d x is

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Q. The value of $\frac{8}{\pi} \int\limits_0^{\frac{\pi}{2}} \frac{(\cos x)^{2023}}{(\sin x)^{2023}+(\cos x)^{2023}} d x$ is

JEE MainJEE Main 2023Integrals

Solution:

$I=\frac{8}{\pi} \int\limits_0^{\frac{\pi}{2}} \frac{(\cos x)^{2023}}{(\sin x)^{2023}+(\cos x)^{2023}} d x$......(1)
Using $\int\limits_0^a f(x) d x=\int\limits_0^a f(a-x) d x$
$I=\frac{8}{\pi} \int\limits_0^{\frac{\pi}{2}} \frac{(\sin x)^{2023}}{(\sin x)^{2023}+(\cos x)^{2023}} d x$........(2)
Adding (1) & (2)
$ 2 I =\frac{8}{\pi} \int\limits_0^{\frac{\pi}{2}} 1 dx$
$ I =2$