Question

The value of $\int \frac{2\sin 2x}{\sqrt{2{\sin }^{2}x+{\cos }^{2}x}}dx$ is
(where $C$ is integration constant)

• A. $2\sqrt{2{\sin }^{2}x+{\cos }^{2}x}+C$
• B. $4\sqrt{2{\sin }^{2}x+{\cos }^{2}x}+C$
• C. $4\sqrt{{\sin }^{2}x+2{\cos }^{2}x}+C$
• D. $2\sqrt{{\sin }^{2}x+2{\cos }^{2}x}+C$

Answer 1

The correct option is B $4\sqrt{2{\sin }^{2}x+{\cos }^{2}x}+C$

Let $I=\int \frac{2\sin 2x}{\sqrt{2{\sin }^{2}x+{\cos }^{2}x}}dx$
Put $2{\sin }^{2}x+{\cos }^{2}x=t,\sin 2xdx=dt$
$\Rightarrow I=2\int t^{-\frac{1}{2}}dt\Rightarrow I=4\sqrt{t}+C=4\sqrt{2{\sin }^{2}x+{\cos }^{2}x}+C$