_0^ x ,f ,( x) ,dx = - Vidyadip

MCQ

$\int_0^\pi x \,f\,(\sin x)\,dx = $

A
$\pi \int_0^\pi {f(\sin x)\,dx} $
✓
$\frac{\pi }{2}\int_0^\pi {f(\sin x)\,dx} $
C
$\frac{\pi }{2}\int_0^{\pi /2} {f(\sin x)\,dx} $
D
None of these

✓

Answer

Correct option: B.

$\frac{\pi }{2}\int_0^\pi {f(\sin x)\,dx} $

b
(b) $\int_0^\pi x f(\sin x)dx = \frac{\pi }{2}\int_0^\pi {f(\sin x)dx} $

Since $\int_0^a {xf(x)dx = \frac{1}{2}a\int_0^a {f(x)dx,} } $

if $f(a - x) = f(x)$.

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