Q. $\int_{}^{}\frac{dx}{\cos{x + \sqrt{3}\sin x}}$ equals
A) $\frac{1}{2}\log{\tan{\left( \frac{x}{2} + \frac{\pi}{12} \right) + c}}$
B) $\frac{1}{2}\log{\tan{\left( \frac{x}{2} - \frac{\pi}{12} \right) + c}}$
C) $\log{\tan{\left( \frac{x}{2} + \frac{\pi}{12} \right) + c}}$
D) $\log{\tan{\left( \frac{x}{2} - \frac{\pi}{12} \right) + c}}$
Explanation
(a)
$\int_{}^{}\frac{dx}{\cos{x + \sqrt{3}\sin x}} = \int_{}^{}\frac{dx}{2\left( \frac{1}{2}\cos{x + \frac{\sqrt{3}}{2}\sin x} \right)}$
$= \frac{1}{2}\int_{}^{}{\sec\left( x - \frac{\pi}{3} \right)}dx$
$= \frac{1}{2}\log{\tan{\left( \frac{x}{2} - \frac{\pi}{6} + \frac{\pi}{4} \right) + c}}$
$= \frac{1}{2}\log{\tan{\left( \frac{x}{2} + \frac{\pi}{12} \right) + c}}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.