The integral int frac{left(1-frac{1}{sqrt{3}} | vedclass

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Question

(A) Let $I = \int \frac{\left(1-\frac{1}{\sqrt{3}}\right)(\cos x-\sin x)}{1+\frac{2}{\sqrt{3}} \sin 2 x} dx = \int \frac{\left(\frac{\sqrt{3}-1}{\sqrt

Vedclass · Accepted Answer

$\frac{1}{2} \log _{ e }\left|\frac{	an \left(\frac{ x }{2}+\frac{\pi}{12}ight)}{	an \left(\frac{ x }{2}+\frac{\pi}{6}ight)}ight|+ C$

Vedclass · Answer

(A) Let $I = \int \frac{\left(1-\frac{1}{\sqrt{3}}ight)(\cos x-\sin x)}{1+\frac{2}{\sqrt{3}} \sin 2 x} dx = \int \frac{\left(\frac{\sqrt{3}-1}{\sqrt{3}}ight)(\cos x-\sin x)}{\frac{\sqrt{3}+2\sin 2x}{\sqrt{3}}} dx = \int \frac{(\sqrt{3}-1)(\cos x-\sin x)}{\sqrt{3}+2\sin 2x} dx$. Multiply numerator and denominator by $\frac{1}{2}$: $I = \int \frac{(\sqrt{3}-1)(\cos x-\sin x)}{2(\frac{\sqrt{3}}{2} + \sin 2x)} dx = \int \frac{(\frac{\sqrt{3}}{2} - \frac{1}{2})(\cos x-\sin x)}{\sin 60^{\circ} + \sin 2x} dx$. Using $\sin 60^{\circ} + \sin 2x = 2 \sin(x + 30^{\circ}) \cos(x - 30^{\circ}) = 2 \sin(x + \frac{\pi}{6}) \cos(x - \frac{\pi}{6})$: $I = \int \frac{(\cos 30^{\circ} - \sin 30^{\circ})(\cos x - \sin x)}{2 \sin(x + \frac{\pi}{6}) \cos(x - \frac{\pi}{6})} dx = \int \frac{\cos(x + 30^{\circ}) - \sin(x - 30^{\circ})}{2 \sin(x + \frac{\pi}{6}) \cos(x - \frac{\pi}{6})} dx$. $I = \frac{1}{2} \int \left( \frac{\cos(x + \frac{\pi}{6})}{\sin(x + \frac{\pi}{6}) \cos(x - \frac{\pi}{6})} - \frac{\sin(x - \frac{\pi}{6})}{\sin(x + \frac{\pi}{6}) \cos(x - \frac{\pi}{6})} ight) dx$. This simplifies to $\frac{1}{2} \int \left( \frac{1}{\sin(x + \frac{\pi}{6})} - \frac{1}{\cos(x - \frac{\pi}{6})} ight) dx$. Using $\int \csc 	heta d	heta = \ln |	an \frac{	heta}{2}|$ and $\int \sec 	heta d	heta = \ln |	an(\frac{	heta}{2} + \frac{\pi}{4})|$: $I = \frac{1}{2} \left( \ln |	an(\frac{x}{2} + \frac{\pi}{12})| - \ln |	an(\frac{x}{2} - \frac{\pi}{12} + \frac{\pi}{4})| ight) + C = \frac{1}{2} \ln \left| \frac{	an(\frac{x}{2} + \frac{\pi}{12})}{	an(\frac{x}{2} + \frac{\pi}{6})} ight| + C$.

The integral $\int \frac{\left(1-\frac{1}{\sqrt{3}}\right)(\cos x-\sin x)}{\left(1+\frac{2}{\sqrt{3}} \sin 2 x\right)} d x$ is equal to

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