First slide

Methods of integration

Question

$\int \frac{dx}{\cos x + \sqrt{3} \sin x}$ is equal to

Moderate

A

$\frac{1}{2} \log \tan (\frac{x}{2} + \frac{π}{12}) + C$
B

$\frac{1}{2} \log \tan (\frac{x}{2} - \frac{π}{12}) + C$
C

$\log \tan (\frac{x}{2} + \frac{π}{12}) + C$
D

$\log \tan (\frac{x}{2} - \frac{π}{12}) + C$

Solution

Now,
$\begin{matrix} \int \frac{dx}{\cos x + \sqrt{3} \sin x} = \int \frac{dx}{2 (\frac{1}{2} \cos x + \frac{\sqrt{3}}{2} \sin x)} \\ = \frac{1}{2} \int \sec (x - \frac{π}{3}) dx \\ = \frac{1}{2} \log \tan (\frac{x}{2} - \frac{π}{6} + \frac{π}{4}) + C \\ = \frac{1}{2} \log \tan (\frac{x}{2} + \frac{π}{12}) + C \end{matrix}$

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