Question

The value of $\int \frac{1}{\sin (x-\frac{\pi }{3})\cos x}dx$ is

• A. $2\log |\sin x+\sin (x-\frac{\pi }{3})|+C$
• B. $2\log |\sin x.\sin (x-\frac{\pi }{3})|+C$
• C. $2\log |\sin x-\sin (x-\frac{\pi }{3})|+C$
• D. None of the above

Answer 1

Answer: (D)

None of the above

Consider, $I=\int \frac{1}{\sin (x-\frac{\pi }{3})\cos x}dx$

$=2\int \frac{\cos (x-\frac{\pi }{3}-x)}{\sin (x-\frac{\pi }{3})\cos x}dx$

$=2\int \frac{\cos (x-\frac{\pi }{3})\cos x+\sin (x-\frac{\pi }{3})\sin x}{\sin (x-\frac{\pi }{3})\cos x}dx$

$=2[\int \frac{\cos (x-\frac{\pi }{3})}{\sin (x-\frac{\pi }{3})}dx+\int \frac{\sin x}{\cos x}dx]$

$=2[\log \left|\sin (x-\frac{\pi }{3})\right|-\log \left|\cos x\right|]+C$

$=2\log |\sec x\cdot \sin (x-\frac{\pi }{3})|+C$