First slide

Evaluation of definite integrals

Question

The value of $\int_{0}^{π / 2} \sqrt{\sin 2 θ} \sin θ d θ$ is

Moderate

A

1
B

0
C

$π / 2$
D

$π / 4$

Solution

$\begin{array}{l} I = \int_{0}^{π / 2} \sqrt{\sin 2 θ} \sin θ d θ \\ = \int_{0}^{π / 2} \sqrt{\sin (π - 2 θ)} \sin (π / 2 - θ) d θ \\ = \int_{0}^{π / 2} \sqrt{\sin 2 θ} \cos θ d θ \\ 2 I = \int_{0}^{π / 2} \sqrt{\sin 2 θ} (\sin θ + \cos θ) d θ \end{array}$
In 2I, put $\sin θ - \cos θ = t$ , so that $t^{2} = 1 - \sin 2 θ \Rightarrow 1 - t^{2} =$
$\sin 2 θ$ . Also $d t = (\cos θ + \sin θ) d θ$
$\begin{array}{l} ∴ 2 I = \int_{- 1}^{1} \sqrt{1 - t^{2}} d t ∣= 2 {[\frac{t}{2} \sqrt{1 - t^{2}} + \frac{1}{2} \sin^{- 1} t]}_{0}^{1} \\ = 2 \frac{π}{4} = \frac{π}{2} \Rightarrow I = \frac{π}{4} . \end{array}$

Get Instant Solutions

When in doubt download our app. Now available Google Play Store- Doubts App

Recieve an sms with download link

Doubts App

OR

SCAN ME

Doubts App

Doubts App