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Definite Integration — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsDefinite Integration2 Marks
MCQ
$\int_0^\pi x f(\sin x) d x$ is equal to
  • A
    $\pi \int_0^\pi f(\cos x) d x$
  • B
    $\pi \int_0^\pi f(\sin x) d x$
  • C
    $\frac{\pi}{2} \int_0^{\frac{\pi}{2}} f (\sin x) d x$
  • ✓
    $\pi \int_0^{\frac{\pi}{2}} f (\cos x) d x$
✓

Answer

Correct option: D.
$\pi \int_0^{\frac{\pi}{2}} f (\cos x) d x$
(D)
Let $I =\int_0^\pi x f (\sin x) d x$ ...(i)
$\therefore \quad I=\int_0^\pi(\pi-x) f(\sin x) d x$ ...(ii)
$\ldots\left[\because \int_0^{ a } f (x) d x=\int_0^{ a } f ( a -x) d x\right]$
Adding (i) and (ii), we get
$2 I=\pi \int_0^\pi f(\sin x) d x$
$\Rightarrow 2 I =2 \pi \int_0^{\frac{\pi}{2}} f (\sin x) d x\left[\begin{array}{c}\because \int_0^{2 a } f (x) d x=2 \int_0^{ a } f (x) d x, \\ \text { if } f (2 a -x)= f (x)\end{array}\right]$
$\Rightarrow I =\pi \int_0^{\frac{\pi}{2}} f (\sin x) d x$
$\Rightarrow I =\pi \int_0^{\frac{\pi}{2}} f \left(\sin \left(\frac{\pi}{2}-x\right)\right) d x$
$\ldots\left[\because \int_0^{ a } f (x) d x=\int_0^{ a } f ( a -x) d x\right]$
$=\pi \int_0^{\frac{\pi}{2}} f (\cos x) d x$

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