QJEEAdv21Ma0185: JEE Advanced 2021 Mathematics - Class 12 JEEAdv21Ma0185 Solution: Let g i : [ π 8 , 3 π 8 ] → R , i = 1 , 2 , and f : [ π 8 , 3 π …

Question

Verified Solution • 2025–26 Session moderate

Let

g_{i} : [\frac{π}{8}, \frac{3 π}{8}] \to R, i = 1, 2

, and

f : [\frac{π}{8}, \frac{3 π}{8}] \to R

be functions such that

g_{1} (x) = 1, g_{2} (x) = | 4 x - π |

and

f (x) = \sin^{2} x

, for all

x \in [\frac{π}{8}, \frac{3 π}{8}]

. Define

S_{i} = \int_{\frac{π}{8}}^{\frac{3 π}{8}} f (x) . g_{i} (x) d x

, i = 1, 2

The value of

\frac{16 S_{1}}{π}

is _____________.

Accepted Answer

S_{1} = \int_{π / 8}^{3 π / 8} \sin^{2} x .1 d x = \int_{π / 8}^{3 π / 8} (\frac{1 - \cos 2 x}{2}) d x

= {(\frac{x}{2} - \frac{\sin 2 x}{4})}_{π / 8}^{3 π / 8} = \frac{π}{8}

∴

\frac{16 S_{1}}{π} = \frac{16}{π} \times \frac{π}{8} = 2

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