Let \(g_i:\left[\frac{\pi}{8}, \frac{3 \pi}{8}\right] \rightarrow \mathbb{R}, i=1,2\), and \(f:\left[\frac{\pi}{8}, \fra…

Let \(g_i:\left[\frac{\pi}{8}, \frac{3 \pi}{8}\right] \rightarrow \mathbb{R}, i=1,2\), and \(f:\left[\frac{\pi}{8}, \frac{3 \pi}{8}\right] \rightarrow \mathbb{R}\) be functions such that \(g_1(x)=1, g_2(x)=|4 x-\pi|\) and \(f(x)=\sin ^2 x\), for all \(x \in\left[\frac{\pi}{8}, \frac{3 \pi}{8}\right]\) Define \( S_i=\int_{\frac{\pi}{8}}^{\frac{3 \pi}{8}} f(x) \cdot g_i(x) d x, i=1,2 \) The value of \(\frac{48 S_2}{\pi^2}\) is