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…

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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, \quad i=1,2 $ The value of 48 S 2 &#960; 2 is ____.

Aicharya · Accepted Answer

S 2 = &#8747; &#960; 8 3 &#960; 8 sin 2 x 4 x - &#960; d x &#160; &#160; &#160; &#8230; &#8230; &#8230; i S 2 = &#8747; &#960; 8 3 &#960; 8 sin 2 3 &#960; 8 + &#960; 8 - x 4 3 &#960; 8 + &#960; 8 - x - &#960; d x S 2 = &#8747; &#960; 8 3 &#960; 8 cos 2 x &#960; - 4 x d x &#160; &#160; &#160; &#160; &#8230; &#8230; &#8230; ii From (i) and (ii), we get 2 S 2 = &#8747; &#960; 8 3 &#960; 8 4 x - &#960; d x 2 &#160; S 2 = &#960; 2 16 S 2 = &#960; 2 32 &#8658; 48 S 2 &#960; 2 = 48 32 &#8658; 48 S 2 &#960; 2 = 3 2 = 1 . 5

Let $g_{i}:\left[\frac{\pi}{8}, \frac{3 \pi}{8}\right] \rightarrow \mathbb{R}, i=1,2$, and…

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