int frac{dx}{sin x+sqrt{3} cos x} equals, whe | vedclass

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

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$\frac{1}{2} \ln \left|	an \left(\frac{x}{2}+\frac{\pi}{6}ight)ight|+c$

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(C) To evaluate the integral $I = \int \frac{dx}{\sin x+\sqrt{3} \cos x}$, we multiply and divide by $2$: $I = \int \frac{dx}{2(\frac{1}{2} \sin x + \frac{\sqrt{3}}{2} \cos x)}$ Using the identity $\sin(x + \frac{\pi}{3}) = \sin x \cos \frac{\pi}{3} + \cos x \sin \frac{\pi}{3} = \frac{1}{2} \sin x + \frac{\sqrt{3}}{2} \cos x$, we get: $I = \frac{1}{2} \int \frac{dx}{\sin(x + \frac{\pi}{3})} = \frac{1}{2} \int \operatorname{cosec}(x + \frac{\pi}{3}) dx$ Using the standard formula $\int \operatorname{cosec} 	heta d	heta = \ln |	an(\frac{	heta}{2})| + c$: $I = \frac{1}{2} \ln |	an(\frac{x + \frac{\pi}{3}}{2})| + c = \frac{1}{2} \ln |	an(\frac{x}{2} + \frac{\pi}{6})| + c$.

$\int \frac{dx}{\sin x+\sqrt{3} \cos x}$ equals, where $c$ is an arbitrary constant.

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