First slide

Evaluation of definite integrals

Question

$\int_{- a}^{a} \sqrt{\frac{a - x}{a + x}} dx$ is equal to

Moderate

A

$π$
B

a
C

$\frac{aπ}{2}$
D

$aπ$

Solution

$\begin{array}{l} I = \int_{- a}^{a} \sqrt{\frac{a - x}{a + x}} dx = \int_{- a}^{a} \frac{a - x}{\sqrt{a^{2} - x^{2}}} dx \\ = a \int_{- a}^{a} \frac{dx}{\sqrt{a^{2} - x^{2}}} - \int_{- a}^{a} \frac{xdx}{\sqrt{a^{2} - x^{2}}} = a \cdot 2 \int_{0}^{a} \frac{dx}{\sqrt{a^{2} - x^{2}}} - 0 (∵ \frac{x}{\sqrt{a^{2} - x^{2}}} is an odd function) \\ = 2 a {[\sin^{- 1} \frac{x}{a}]}_{0}^{a} \\ \Rightarrow 2 a [\sin^{- 1} (1) - \sin^{- 1} (0)] = πa \end{array}$

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