Question

$\int \frac{dx}{\sin x+\sqrt{3}\cos x}=?$

• A. $\frac{1}{4}ln|\sec (x-\frac{\pi }{6})+\tan (x-\frac{\pi }{6})|+C$
• B. $ln|\sec (x-\frac{\pi }{6})+\tan (x-\frac{\pi }{6})|+C$
• C. $\frac{1}{2}\ln |\sec (x+\frac{\pi }{6})+\tan (x+\frac{\pi }{6})|+C$
• D. $\frac{1}{2}ln|\sec (x-\frac{\pi }{6})+\tan (x-\frac{\pi }{6})|+C$

Answer 1

The correct option is D $\frac{1}{2}ln|\sec (x-\frac{\pi }{6})+\tan (x-\frac{\pi }{6})|+C$

$\begin{array}{c}\int \frac{dx}{sinx+\sqrt{3}cosx}\\ \Rightarrow \int \frac{dx}{2\left(\frac{1}{2}sinx+\frac{\sqrt{3}}{2}cosx\right)}\\ \Rightarrow \frac{1}{2}\int \frac{dx}{sinx\cdot \sin {30}^0+cos{30}^0\cdot \cos x}\\ \Rightarrow \frac{1}{2}\int \frac{dx}{\cos \left(x-\frac{\pi }{6}\right)}\\ \Rightarrow \frac{1}{2}\int \sec \left(x-\frac{\pi }{6}\right)dc\\ \Rightarrow \frac{1}{2}In\left|\sec \left(x-\frac{\pi }{6}\right)+\tan \left(x-\frac{\pi }{6}\right)\right|+C\end{array}$