Let g_ i :[(π)/(8), (3 π)/(8)] arrow ℝ, i=1,2 , and f:[(π)/(8), (3 π)/(8)] arrow ℝ be functions such that g_ 1 (x)=1, g_ 2 (x)=|4 x-π| and f(x)= ^ 2 x , for all x ∈[(π)/(8), (3 π)/(8)] Define S_ i =∫_ (π)/(8) ^ (3

Correct answer: 0 S _ 1 = ∫ _ (π)/(8) ^ (3π)/(8) ^ 2 x · 1dx . . . (i) S _ 1 = ∫ _ (π)/(8) ^ (3π)/(8) ^ 2 ((3π)/(8) + (π)/(8) - x)dx S _ 1 = ∫ _ (π)/(8) ^ (3π)/(8) ^ 2 xdx . . (ii) From (i) and (ii), we get 2 S _ 1 = ∫ _ (π)/(8) ^ (3π)/(8) 1dx 2 S _ 1 = (π)/(4) 16 S _ 1 π = 2

JEE Advanced 2021Paper 2

Exam: JEE Advanced
Year: 2021
Paper: Paper 2
Subject: Mathematics
Chapter: Definite Integration

JEE Advanced 2021 · Paper 2 · Mathematics · Definite Integration

Let g_ i :[(π)/(8), (3 π)/(8)] arrow ℝ, i=1,2 , and f:[(π)/(8), (3...

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

\dfrac{16{S}_{1}}{\pi}

is ___.

Correct answer

Step-by-step solution

${S}_{1} = {∫}_{\dfrac{\pi}{8}}^{\dfrac{3\pi}{8}}\sin^{2}x \cdot 1dx \ldots \ldots . . . \left(i\right)$

${S}_{1} = {∫}_{\dfrac{\pi}{8}}^{\dfrac{3\pi}{8}}\sin^{2}\left(\dfrac{3\pi}{8} + \dfrac{\pi}{8} - x\right)dx$

${S}_{1} = {∫}_{\dfrac{\pi}{8}}^{\dfrac{3\pi}{8}}\cos^{2}xdx \ldots \ldots . . \left(ii\right)$

From (i) and (ii), we get

$2{S}_{1} = {∫}_{\dfrac{\pi}{8}}^{\dfrac{3\pi}{8}}1dx$

$2{S}_{1} = \dfrac{\pi}{4}$

$\dfrac{16{S}_{1}}{\pi} = 2$

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