Question

$\int \sqrt{1+\cos x}dx$ is equal to

• A. $2\sin \left(\frac{x}{2}\right)+C$
• B. $\sqrt{2}\sin \left(\frac{x}{2}\right)+C$
• C. $\frac{1}{2}\sin \left(\frac{x}{2}\right)+C$
• D. $\frac{\sqrt{2}}{2}\sin \left(\frac{x}{2}\right)+C$
• E. $2\sqrt{2}\sin \left(\frac{x}{2}\right)+C$

Answer 1

Answer: (E)

Let $I=\int \sqrt{1+\cos x}dx=\int \sqrt{2{\cos }^2\frac{x}{2}}dx$
$=\int \sqrt{2}\cos \frac{x}{2}dx$
$=\frac{\sqrt{2}\sin \frac{x}{2}}{1/2}+C$
$=2\sqrt{2}\sin \left(\frac{x}{2}\right)+C$