$\int \frac{d x}{\cos x+\sqrt{3} \sin x}$ 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) $\log \tan \left(\frac{x}{2}-\frac{\pi}{12}\right)+c$

JEE Main · 2007 · 29 Apr 2007 · Offline · Indefinite Integrals

\int \frac{d x}{c o s x + 3 s i n x}

equals

Answer: (A)

Write the denominator as a single cosine,

cos x + 3 sin x = 2 cos (x - \frac{π}{3})

. The integral then becomes

\frac{1}{2} \int sec (x - \frac{π}{3}) d x

, and using

\int sec u d u = lo g tan (\frac{u}{2} + \frac{π}{4})

this is

\frac{1}{2} lo g tan (\frac{x}{2} - \frac{π}{6} + \frac{π}{4}) + c = \frac{1}{2} lo g tan (\frac{x}{2} + \frac{π}{12}) + c

JEE Main · 2007 · 29 Apr 2007 · Offline · Single correct option