First slide

Evaluation of definite integrals

Question

$Evaluate \int_{π / 6}^{π / 3} \frac{\sqrt{(\sin x)} d x}{\sqrt{(\sin x)} + \sqrt{(\cos x)}}$

Moderate

A

$\frac{π}{2}$
B

$\frac{π}{12}$
C

$\frac{π}{3}$
D

$\frac{π}{4}$

Solution

$Given integral I = \int_{π / 6}^{π / 3} \frac{\sqrt{(\sin x)} d x}{\sqrt{(\sin x)} + \sqrt{(\cos x)}} - - - - - - (1)$
$or I = \int_{π / 6}^{π / 3} \frac{\sqrt{(\cos x)} d x}{\sqrt{(\cos x)} + \sqrt{(\sin x)}} (Replacing x by \frac{π}{2} - x) - - - (2)$
Adding equations (l ) and (2), we get
$\begin{array}{l} 2 I = \int_{π / 6}^{π / 3} \frac{\sqrt{(\sin x)} + \sqrt{(\cos x)}}{\sqrt{(\cos x)} + \sqrt{(\sin x)}} d x \\ = \int_{π / 6}^{π / 3} d x = [x]_{π / 6}^{π / 3} = π / 3 - π / 6 = π / 6 \\ ⇒ I = π / 12 \end{array}$

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