Question

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

• A. $\frac{1}{2}{\log }_e\left|\frac{\tan \left(\frac{x}{2}+\frac{\pi }{12}\right)}{\left(\frac{x}{2}+\frac{\pi }{6}\right)}\right|+C$
• B. $\frac{1}{2}{\log }_e\left|\frac{\tan \left(\frac{x}{2}+\frac{\pi }{6}\right)}{\left(\frac{x}{2}+\frac{\pi }{3}\right)}\right|+C$
• C. ${\log }_e\left|\frac{\tan \left(\frac{x}{2}+\frac{\pi }{6}\right)}{\tan \left(\frac{x}{2}+\frac{\pi }{12}\right)}\right|+C$
• D. $\frac{1}{2}{\log }_e\left|\frac{\tan \left(\frac{x}{2}-\frac{\pi }{12}\right)}{\tan \left(\frac{x}{2}-\frac{\pi }{6}\right)}\right|+C$

Answer 1

Answer: (A)

$I=\int \frac{\left(1-\frac{1}{\sqrt{3}}\right)(\cos x-\sin x)}{\left(1+\frac{2}{\sqrt{3}}\sin 2x\right)}dx$
$\frac{\sqrt{3}}{2}\int \frac{\left(1-\frac{1}{\sqrt{3}}\right)(\cos x-\sin x)}{\left(\frac{\sqrt{3}}{2}+\sin 2x\right)}dx$
$\int \frac{\left(\frac{\sqrt{3}}{2}-\frac{1}{2}\right)(\cos x-\sin x)}{\sin 60^{\circ }+\sin 2x}dx$
$\int \frac{\left(\frac{\sqrt{3}}{2}\cos x-\frac{1}{2}\cos x-\frac{\sqrt{3}}{2}\sin x+\frac{1}{2}\sin x\right)}{2\sin \left(x+\frac{\pi }{6}\right)\cos \left(x-\frac{\pi }{6}\right)}dx$
$\int \frac{\left(\cos \left(x-\frac{\pi }{6}\right)-\sin \left(x+\frac{\pi }{6}\right)\right)}{2\sin \left(x+\frac{\pi }{6}\right)\cos \left(x-\frac{\pi }{6}\right)}dx$
$\frac{1}{2}\left(\int \frac{dx}{\sin \left(x+\frac{\pi }{6}\right)}-\int \frac{dx}{\cos \left(x-\frac{\pi }{6}\right)}\right)$
$\frac{1}{2}\ln \left|\frac{\tan \left(\frac{x}{2}+\frac{\pi }{12}\right)}{\tan \left(\frac{x}{2}+\frac{\pi }{6}\right)}\right|$