Question

\[\int\frac{1}{\cos x + \sqrt{3} \sin x} \text{ dx } \] is equal to

Options

• `  log   tan (x/3  + π / 2) + C `

• \[\text{ log  tan}   \left( \frac{x}{2} - \frac{\pi}{3} \right) + C\]

• `   1/2  log   tan (x/2  + π /3 ) + C `

• none of these

Answer 1

 none of these

 

\[\int\frac{1}{\cos x + \sqrt{3}\sin x}dx\]
\[ = \frac{1}{2}\int\frac{dx}{\cos x \times \frac{1}{2} + \sin x \times \frac{\sqrt{3}}{2}}\]
\[ = \frac{1}{2}\int\frac{dx}{\cos x \cdot \cos\frac{\pi}{3} + \sin x \cdot \sin\frac{\pi}{3}}\]
\[ = \frac{1}{2}\int\frac{dx}{\cos \left( x - \frac{\pi}{3} \right)}\]
\[ = \frac{1}{2}\int\sec \left( x - \frac{\pi}{3} \right)dx\]
\[ = \frac{1}{2}\text{ ln }\left| \tan \left\{ \frac{\pi}{4} + \frac{1}{2}\left( x - \frac{\pi}{3} \right) \right\} \right| + C\]
\[ = \frac{1}{2}\text{ ln  }\left| \tan \left( \frac{\pi}{4} + \frac{x}{2} - \frac{\pi}{6} \right) \right| + C\]
\[ = \frac{1}{2}\text{ ln }\left| \tan \left( \frac{x}{2} + \frac{\pi}{12} \right) \right| + C\]