Question

Integrate the following function: $\frac{\cos x}{\sqrt{1+\sin x}}$

• A. $\sqrt{2}\sin (\frac{\pi }{4}+\frac{x}{2})+c$
• B. $\sin (\frac{\pi }{4}+\frac{x}{2})+c$
• C. $\frac{1}{\sqrt{2}}\sin (\frac{\pi }{4}+\frac{x}{2})+c$
• D. $2\sqrt{2}\sin (\frac{\pi }{4}+\frac{x}{2})+c$

Answer 1

The correct option is D $2\sqrt{2}\sin (\frac{\pi }{4}+\frac{x}{2})+c$

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

$\because \sin x=2\sin \frac{x}{2}\cos \frac{x}{2}$

$\cos x={\cos }^2\frac{x}{2}-{\sin }^2\frac{x}{2}$

Then, we have

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

$=\int \frac{({\cos }^2\frac{x}{2}-{\sin }^2\frac{x}{2})}{(\cos \frac{x}{2}+\sin \frac{x}{2})}dx$

$=\int \frac{(\cos \frac{x}{2}-\sin \frac{x}{2})(\cos \frac{x}{2}+\sin \frac{x}{2})}{(\cos \frac{x}{2}+\sin \frac{x}{2})}dx$

$=\int (\cos \frac{x}{2}-\sin \frac{x}{2})dx$

$=\int \cos \frac{x}{2}.dx-\int \sin \frac{x}{2}.dx$

$=2\sin \frac{x}{2}+2\cos \frac{x}{2}+c$

$=2.\sqrt{2}\sin (\frac{\pi }{4}+\frac{x}{2})+c$