Question

If $\int \sqrt{1+\sin \frac{x}{2}}dx=A\sin \left(\frac{x}{4}-\frac{\pi }{4}\right)$ then value of $A$ is

• A. $2\sqrt{2}$
• B. $\sqrt{2}$
• C. $\frac{1}{\sqrt{2}}$
• D. $4\sqrt{2}$

Answer 1

The correct option is D $4\sqrt{2}$

Consider,

$\frac{dA\sin (\frac{x}{4}-\frac{\pi }{4})}{dx}=\frac{A}{4}\cos (\frac{x}{4}-\frac{\pi }{4})$

We know,

$\cos \left(2(\frac{x}{4}-\frac{\pi }{4})\right)=2{\cos }^2(\frac{x}{4}-\frac{\pi }{4})-1=\cos (\frac{x}{2}-\frac{\pi }{2})$

$\therefore \sqrt{2}\cos (\frac{x}{4}-\frac{\pi }{4})=\sqrt{\cos (\frac{x}{2}-\frac{\pi }{2})+1}=\sqrt{\sin \frac{x}{2}+1}$

So, $\frac{A}{4}=\sqrt{2}$

$\Rightarrow A=4\sqrt{2}$