int frac{1}{cos x+sqrt{3} sin x} dx = | vedclass

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(A) To evaluate the integral $I = \int \frac{1}{\cos x+\sqrt{3} \sin x} dx$,we multiply and divide the denominator by $2$: $I = \int \frac{1}{2(\frac{

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$\frac{1}{2} \log \left|	an \left(\frac{x}{2}+\frac{\pi}{12}ight)ight|+c$

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(A) To evaluate the integral $I = \int \frac{1}{\cos x+\sqrt{3} \sin x} dx$,we multiply and divide the denominator by $2$: $I = \int \frac{1}{2(\frac{1}{2} \cos x + \frac{\sqrt{3}}{2} \sin x)} dx$ Using the identity $\sin(A+B) = \sin A \cos B + \cos A \sin B$,we can write $\frac{1}{2} \cos x + \frac{\sqrt{3}}{2} \sin x = \sin(x + \frac{\pi}{6})$: $I = \frac{1}{2} \int \frac{1}{\sin(x + \frac{\pi}{6})} dx$ $I = \frac{1}{2} \int \csc(x + \frac{\pi}{6}) dx$ Using the standard integral formula $\int \csc 	heta d	heta = \log |	an(\frac{	heta}{2})| + C$: $I = \frac{1}{2} \log |	an(\frac{x}{2} + \frac{\pi}{12})| + C$

$\int \frac{1}{\cos x+\sqrt{3} \sin x} dx =$

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