Question

Solve: $\int \frac{dx}{\cos x\sqrt{1+\cos 2x+\sin 2x}};(0<x<\frac{\pi }{4})$

• A. $2+\sqrt{\cot x}$
• B. $\sqrt{\tan x+1}$
• C. $\sqrt{2+2\tan x}$
• D. $\sqrt{2+2\cot x}$

Answer 1

Answer: (C)

$\sqrt{2+2\tan x}$

Consider, $I=\int \frac{dx}{\cos x\sqrt{1+\cos 2x+\sin 2x}}$

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

$I=\int \frac{dx}{\cos x\sqrt{2{\cos }^{2}x+2\sin x\cos x}}$

$I=\int \frac{{\sec }^{2}xdx}{\sqrt{2+2\tan x}}$

Put $\tan x=t$

Then ${\sec }^{2}xdx=dt$

Therefore, $I=\frac{dt}{\sqrt{2+2t}}$

Thus $\sqrt{2+2\tan x}+C$