_0^ /2 ^4 x ^6 x ,dx equals - Vidyadip

MCQ

$\int_0^{\pi /2} {{{\sin }^4}x{{\cos }^6}x\,dx} $ equals

A
$\frac{{5\pi }}{{512}}$
✓
$\frac{{3\pi }}{{512}}$
C
$\frac{\pi }{{512}}$
D
None of these

✓

Answer

Correct option: B.

$\frac{{3\pi }}{{512}}$

b
(b) $I = \int_0^{\pi /2} {{{\sin }^4}x{{\cos }^6}x.dx} $

$ \Rightarrow I = \frac{{\Gamma \,(5/2)\,\Gamma \,(7/2)}}{{2\Gamma (6)}}$, (Applying gamma function)

==> $I = \frac{{3/2.1/2.\sqrt \pi .5/2.3/2.1/2.\sqrt \pi }}{{2.5.4.3.2.1}}$

$ = \frac{{3\pi }}{{512}}$.

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