Question

If $I=\int (\sqrt{\cot x}-\sqrt{\tan x})dx$, then $I$ equals

• A. $\sqrt{2}\log (\sqrt{\tan x}-\sqrt{\cot x})+C$
• B. $\sqrt{2}\log |\sin x+\cos x+\sqrt{\sin 2x}|+C$
• C. $\sqrt{2}\log |\sin x-\cos x+\sqrt{2}\sin x\cos x|+C$
• D. $\sqrt{2}\log |\sin (x+\pi /4)+\sqrt{2}\sin x\cos x|+C$

Answer 1

Answer: (B)

$\sqrt{2}\log |\sin x+\cos x+\sqrt{\sin 2x}|+C$

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