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STD 12 - 7.2 definite integral — JEE STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceJEESTD 12 - 7.2 definite integral4 Marks
MCQ
Let ${I_1} = \int_a^{\pi - a} {xf(\sin x)dx,\,{I_2} = \int_a^{\pi - a} {\,\,f(\sin x)dx} } $, then ${I_2}$ is equal to
  • A
    $\frac{\pi }{2}{I_1}$
  • B
    $\pi \,{I_1}$
  • ✓
    $\frac{2}{\pi }{I_1}$
  • D
    $2{I_1}$
✓

Answer

Correct option: C.
$\frac{2}{\pi }{I_1}$
c
(c) ${I_1} = \int_a^{\pi - a} {xf(\sin x)dx} $

$ = \int_a^{\pi - a} {(\pi - x)\,f\,(\sin (\pi - x))\,dx} $,

$[  \because \int_a^b {f(x)dx = \int_a^b {f(a + b - x)\,dx} } ]$

$ = \int_a^{\pi - a} {(\pi - x)\,f\,(\sin x)\,dx} $

$ = \int_a^{\pi - a} {\pi \,f\,(\sin x)\,dx - {I_1}} $

$ \Rightarrow 2{I_1} = \pi \,{I_2}\, $

$\Rightarrow {I_2} = \frac{2}{\pi }{I_1}$.

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