Q. $\int_{}^{}\sqrt{1 + \sin\frac{x}{2}}\ dx$ is equal to
A) $\frac{1}{4}\left\lbrack \cos{\frac{x}{4} - \sin\frac{x}{4}} \right\rbrack + c$
B) $4\left\lbrack \cos{\frac{x}{4} - \sin\frac{x}{4}} \right\rbrack + c$
C) $4\left\lbrack \sin{\frac{x}{4} - \cos\frac{x}{4}} \right\rbrack + c$
D) $4\left\lbrack \sin{\frac{x}{4} + \cos\frac{x}{4}} \right\rbrack + c$
Explanation
(c)
$\int_{}^{}\sqrt{1 + \sin\frac{x}{2}}\ dx$
$= \int_{}^{}\sqrt{\sin^{2}{\frac{x}{4} + \cos^{2}\frac{x}{4}} + 2\sin{\frac{x}{4}\cos\frac{x}{4}}}\ dx$
$= \int_{}^{}\left\lbrack \sin{\frac{x}{4} + \cos\frac{x}{4}} \right\rbrack dx$
$= 4\left( \sin{\frac{x}{4} - \cos\frac{x}{4}} \right) + c$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.