_ - /2 ^ /2 ^2 x ^2 x( x + x) ,dx = - Vidyadip

MCQ

$\int_{ - \pi /2}^{\pi /2} {{{\sin }^2}x{{\cos }^2}x(\sin x + \cos x)\,dx = } $

A
$\frac{2}{{15}}$
✓
$\frac{4}{{15}}$
C
$\frac{6}{{15}}$
D
$\frac{8}{{15}}$

✓

Answer

Correct option: B.

$\frac{4}{{15}}$

b
(b) $\int_{ - \pi /2}^{\pi /2} {{{\sin }^2}x{{\cos }^2}x(\sin x + \cos x)dx} $

$= \int_{ - \pi /2}^{\pi /2} {{{\sin }^3}x{{\cos }^2}xdx + \int_{ - \pi /2}^{\pi /2} {{{\sin }^2}x{{\cos }^3}x\,dx} } $

$ = 0 + 2\int_0^{\pi /2} {{{\sin }^2}x{{\cos }^3}xdx} $

$ = 0 + 2 \times \frac{2}{{15}} = \frac{4}{{15}}$ .

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