Question bank/JEE Advanced/Mathematics

Let $\mathrm{g}_i:\left[\frac{\pi}{8}, \frac{3 \pi}{8}\right] \rightarrow \mathbb{R}, \mathrm{i}=1,2$, and $f:\left[\frac{\pi}{8}, \frac{3 \pi}{8}\right] \rightarrow \mathbb{R}$ be functions such that $\mathrm{g}_1(\mathrm{x})=1, \mathrm{~g}_2(\mathrm{x})=|4 \mathrm{x}-\pi|$ and $f(\mathrm{x})=\sin ^2 \mathrm{x}$, for all $\mathrm{x} \in\left[\frac{\pi}{8}, \frac{3 \pi}{8}\right]$
Define $\mathrm{S}_i=\int_{\frac{\pi}{8}}^{\frac{3 \pi}{8}} f(\mathrm{x}) \cdot \mathrm{g}_i(\mathrm{x}) \mathrm{dx}, i=1,2$. What is the value of $\frac{16 S_{1}}{\pi}$?

Source: JEEBench (IIT Delhi) / JEE Adv 2021 Paper 2