Psi Integral! (12)

$\large \displaystyle \text{Let } I = \int_0^{\frac{\pi}{2}} \frac{\psi (x)}{\psi (x) + \psi \left(\frac{\pi}{2} - x \right)} \, dx\\ \large \displaystyle \left[\because \int_0^a f(x) \, dx = \int_0^a f(x-a) \, dx \right]\\ \large \displaystyle \therefore I = \int_0^{\frac{\pi}{2}} \frac{\psi \left(\frac{\pi}{2} - x \right)}{\psi \left(\frac{\pi}{2} - x \right) + \psi \left(\frac{\pi}{2} - \frac{\pi}{2} + x \right)} \, dx\\ \large \displaystyle \color{#69047E}{\text{By adding the above two equations}}\\ \large \displaystyle \implies 2I = \int_0^{\frac{\pi}{2}} \frac{\psi \left(\frac{\pi}{2} - x \right) + \psi(x)}{\psi \left(\frac{\pi}{2} - x \right) + \psi (x)} \, dx\\ \large \displaystyle \implies 2I= \int_0^{\frac{\pi}{2}} \, dx\\ \large \displaystyle \implies 2I = \left[x \right]_0^{\frac{\pi}{2}}\\ \large \displaystyle \implies 2I = \frac{\pi}{2} - 0\\ \large \displaystyle \implies I = \frac{\pi}{4} \approx \color{#3D99F6}{\boxed{0.785714}}\\ \large \displaystyle \implies A = \color{#20A900}{0.785714}\\ \large \displaystyle \implies \lfloor 1000 \times A \rfloor = \lfloor 0.785714 \times 1000 \rfloor = \lfloor 785.714 \rfloor = \color{#D61F06}{\boxed{785}}$