Question

If $I=\int (\sqrt{\cot x}-\sqrt{\tan x})dx$ equals $\sqrt{2}\log |f\left(x\right)+\sqrt{g\left(x\right)}|+C$, then which of the following is/are correct ?

• A. $f\left(x\right)=\sin x+\cos x$
• B. $g\left(x\right)=\sin 2x$
• C. $g\left(x\right)=\sin x$
• D. $f\left(x\right)=\sin x-\cos x$

Answer 1

Answer: (A), (B)

$g\left(x\right)=\sin 2x$

$I=\int \frac{\cos x-\sin x}{\sqrt{\cos x\sin x}}dx$
Put $\sin x+\cos x=t$, so that $2\sin x\cos x=t^2-1$
$\therefore I=\sqrt{2}\int \frac{dt}{\sqrt{t^2-1}}=\sqrt{2}\log |t+\sqrt{t^2-1}|+C$
$=\sqrt{2}\log |\sin x+\cos x+\sqrt{\sin 2x}|+C$