The value of 8 π ∫ 0 π 2 cos x 2023 sin x 2023 + cos x 2023 d x is ______.

Let

$I=\frac{8}{\pi }{\int }_0^{\frac{\pi }{2}}\frac{{\left(\cos x\right)}^{2023}}{{\left(\sin x\right)}^{2023}+{\left(\cos x\right)}^{2023}}dx ...\left(1\right)$

Using ${\int }_0^{a}f\left(x\right)\mathrm{dx}={\int }_0^{a}f(a-x)\mathrm{dx}$

$I=\frac{8}{\pi }{\int }_0^{\frac{\pi }{2}}\frac{{\left(\sin x\right)}^{2023}}{{\left(\sin x\right)}^{2023}+{\left(\cos x\right)}^{2023}}dx ...\left(2\right)$

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

$2I=\frac{8}{\pi }{\int }_0^{\frac{\pi }{2}}1 \mathrm{dx}$

$\Rightarrow 2I=\frac{8}{\mathrm{\pi }}{\left[x\right]}_0^{\frac{\mathrm{\pi }}{2}}$

$\Rightarrow I=2$