Question

If $I=\int (\sqrt{\frac{a+x}{a-x}}+\sqrt{\frac{a-x}{a+x}})dx$, then I equals

• 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 $2a{\sin }^{-1}\left(\frac{x}{a}\right)+C$

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

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

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

We know that, $\int \frac{1}{\sqrt{a^2-x^2}}dx={\sin }^{-1}\left(\frac{x}{a}\right)+c$

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