First slide

Evaluation of definite integrals

Question

$\int_{0}^{π / 2} \frac{\sqrt{\sin . x}}{\sqrt{\sin x} + \sqrt{\cos x}} d x =$

Moderate

A

$\frac{π}{4}$
B

$\frac{π}{2}$
C

0
D

1

Solution

$I = \int_{0}^{π / 2} \frac{\sqrt{\sin x}}{\sqrt{\sin x} + \sqrt{\cos x}} d x - - - - - (1)$
$I = \int_{0}^{π / 2} \frac{\sqrt{\sin (\frac{π}{2} - x)}}{\sqrt{\sin (\frac{π}{2} - x)} + \sqrt{\cos (\frac{π}{2} - x)}} d x = \int_{0}^{π / 2} \frac{\sqrt{\cos x}}{\sqrt{\cos x} + \sqrt{\sin x}} d x - - - - - (2)$
$\begin{array}{l} (1) + (2) \Rightarrow 2 I = \int_{0}^{π / 2} (1) d x = (x)_{0}^{π / 2} = \frac{π}{2} \\ I = \frac{π}{4} \end{array}$

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