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\int \frac{dx}{\cos x+\sqrt{3}\sin x}\; \; equals

  • Option 1)

    \log \tan \left ( \frac{x}{2}+\frac{\pi }{12} \right )+C\; \;

  • Option 2)

    \log \tan \left ( \frac{x}{2}-\frac{\pi }{12} \right )+C\;

  • Option 3)

    \; \frac{1}{2}\log \tan \left ( \frac{x}{2}+\frac{\pi }{12} \right )+C

  • Option 4)

    \; \frac{1}{2}\log \tan \left ( \frac{x}{2}-\frac{\pi }{12} \right )+C

 

Answers (1)

best_answer

As the learnt in concept

Integrals for Trigonometric functions -

\frac{\mathrm{d} }{\mathrm{d} x}\left ( -cos x \right ) =sinx

\therefore \int sinxdx=-cosx+c

-

 

I=\int \frac{dx}{cosx+\sqrt{3}sinx}dx

{cosx+\sqrt{3}sinx} = 2cos\left (x-\frac{\pi}{3} \right )

I=\frac{1}{2}\int \frac{dx}{\left (cosx-\frac{\pi}{3} \right )}dx

I=\frac{1}{2}\int sec{\left (x-\frac{\pi}{3} \right )}dx

I= \frac{1}{2}log\: tan\left (\frac{x}{2}-\frac{\pi}{6}+\frac{\pi}{4} \right )+C

I= \frac{1}{2}log\: tan\left (\frac{x}{2}+\frac{\pi}{12} \right ) +C


Option 1)

\log \tan \left ( \frac{x}{2}+\frac{\pi }{12} \right )+C\; \;

Incorrect

Option 2)

\log \tan \left ( \frac{x}{2}-\frac{\pi }{12} \right )+C\;

Incorrect

Option 3)

\; \frac{1}{2}\log \tan \left ( \frac{x}{2}+\frac{\pi }{12} \right )+C

Correct

Option 4)

\; \frac{1}{2}\log \tan \left ( \frac{x}{2}-\frac{\pi }{12} \right )+C

Incorrect

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