Single Choice

$$\int \dfrac {dx}{\cos x + \sqrt {3}\sin x}$$ equals

A$$\dfrac {1}{2}\log \tan \left (\dfrac {x}{2} + \dfrac {\pi}{12}\right ) + C$$
Correct Answer
B13logtan(x2−π12)+C
Clogtan(x2+π6)+C
D12logtan(x2−π6)+C

Solution

$$\int \dfrac {dx}{\cos x + \sqrt {3}\sin x}$$
$$= \dfrac {1}{2}\int \dfrac {dx}{\dfrac {1}{2}\cos x + \dfrac {\sqrt {3}}{2}\sin x}$$
$$= \dfrac {1}{2}\int \dfrac {dx}{\cos \dfrac {\pi}{3} \cos x + \sin \dfrac {\pi}{3} \sin x}$$
$$= \dfrac {1}{2}\int \dfrac {dx}{\cos \left (x - \dfrac {\pi}{3}\right )}$$
$$= \dfrac {1}{2} \int \sec \left (x - \dfrac {\pi}{3}\right )dx$$
$$= \dfrac {1}{2}\log \tan \left (\dfrac {x}{2} - \dfrac {\pi}{6} + \dfrac {\pi}{4}\right ) + C$$
$$= \dfrac {1}{2}\log \tan \left (\dfrac {x}{2} + \dfrac {\pi}{12}\right ) + C$$
Hence, option A is correct.

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