Mathematics - Definite Integration Question with Solution | TestHub

Let $g_i:\left[\frac{\pi }{8},\frac{3\pi }{8}\right]\to \mathbb{R},i=1,2$, and $f:\left[\frac{\pi }{8},\frac{3\pi }{8}\right]\to \mathbb{R}$ be functions such that $g_1(x)=1,g_2(x)=|4x-\pi |$ and $f(x)={\sin }^{2}x$, for all $x\in \left[\frac{\pi }{8},\frac{3\pi }{8}\right]$ Define $$S_i={\int }_{\frac{\pi }{8}}^{\frac{3\pi }{8}}f(x)\cdot g_i(x)dx,i=1,2$$ The value of$\frac{16S_1}{\pi }$is ___.