The correct evaluation of _0^ /2 x , 2x is - Vidyadip

MCQ

The correct evaluation of $\int_0^{\pi /2} {\sin x\,\sin 2x} $ is

A
$\frac{4}{3}$
B
$\frac{1}{3}$
C
$\frac{3}{4}$
✓
$\frac{2}{3}$

✓

Answer

Correct option: D.

$\frac{2}{3}$

d
(d) Let $I = \int_0^{\pi /2} {\sin x\sin 2x\,dx} $

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

Put $t = \sin x \Rightarrow dt = \cos x\,dx$

Now, $I = 2\int_0^1 {{t^2}dt = \frac{2}{3}[{t^3}]_0^1 = \frac{2}{3}} $.

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