Question

If $I=\int \sqrt{\frac{5-x}{5+x}}dx$, then $I$ equals

• A. $5{\sin }^{-1}\left(\frac{x}{5}\right)+\sqrt{25-x^2}+C$
• B. $10{\sin }^{-1}\left(\frac{x}{5}\right)+\sqrt{25-x^2}+C$
• C. $5{\sin }^{-1}\left(\frac{x}{5}\right)-\sqrt{25-x^2}+C$
• D. none of these

Answer 1

The correct option is A $5{\sin }^{-1}\left(\frac{x}{5}\right)+\sqrt{25-x^2}+C$

$I=\int \sqrt{\frac{5-x}{5+x}}dx$
$\Rightarrow I=\int \frac{5-x}{\sqrt{25-x^2}}dx$
$=\int \frac{5}{\sqrt{25-x^2}}dx-\int \frac{x}{\sqrt{25-x^2}}dx$
Put $25-x^2=t$ in the second integral
$\Rightarrow xdx=-\frac{dt}{2}$
$=5{\sin }^{-1}\left(\frac{x}{5}\right)+\frac{1}{2}\int \frac{dt}{\sqrt{t}}$
$=5{\sin }^{-1}\left(\frac{x}{5}\right)+\sqrt{t}+C$
$\Rightarrow I=5{\sin }^{-1}\left(\frac{x}{5}\right)+\sqrt{25-x^2}+C$