Question

Evaluate : $\int (\sqrt{\frac{a+x}{a-x}}+\sqrt{\frac{a-x}{a+x}})dx$

• A. $2{\sin }^{-1}\left(\frac{x}{a}\right)+c$
• B. $2a{\sin }^{-1}\left(\frac{x}{a}\right)+c$
• C. $2{\cos }^{-1}\left(\frac{x}{a}\right)+c$
• D. $2a{\cos }^{-1}\left(\frac{x}{a}\right)+c$

Answer 1

The correct option is B $2a{\sin }^{-1}\left(\frac{x}{a}\right)+c$

Given, $I=$ $\int \frac{\sqrt{a-x}}{\sqrt{a+x}}+\frac{\sqrt{a+x}}{\sqrt{a-x}}dx$

$=\int \frac{(a+x)+(a-x)}{\sqrt{a^2-x^2}}dx$

$=2\int \frac{a}{\sqrt{a^2-x^2}}dx$

$=2.{\sin }^{-1}\frac{x}{a}.\left(a\right)+c$

$=2a.{\sin }^{-1}\frac{x}{a}+c$