The value of (8)/(π) ∫ _ 0 ^ (π)/(2) ( x ) ^ 2023 ( x) ^ 2023 + (...

Let

I = \dfrac{8}{\pi}{∫}_{0}^{\dfrac{\pi}{2}}\dfrac{{\left(\cos x\right)}^{2023}}{{\left(\sin x\right)}^{2023} + {\left(\cos x\right)}^{2023}}dx . . . \left(1\right)$</p><p>Using${∫}_{0}^{a}f ( x ) dx = {∫}_{0}^{a}f ( a - x ) dx$</p><p>$I = \dfrac{8}{\pi}{∫}_{0}^{\dfrac{\pi}{2}}\dfrac{{\left(\sin x\right)}^{2023}}{{\left(\sin x\right)}^{2023} + {\left(\cos x\right)}^{2023}}dx . . . \left(2\right)

Adding \left(1\right)$and \left(2\right)$, we get

$2I = \dfrac{8}{\pi}{∫}_{0}^{\dfrac{\pi}{2}}1dx$

$⇒ 2I = \dfrac{8}{\pi}{\left[x\right]}_{0}^{\dfrac{\pi}{2}}$

$⇒ I = 2$