Question

The value of the integral \(\int \frac{1}{\sin x-\sqrt{3}\cos x}dx\) is....

• A. \(\frac{1}{2}\log |\csc \left(x-\frac{\pi }{3}\right)+\cot \left(x-\frac{\pi }{3}\right)|+c\)
• B. \(\frac{1}{2}\log |\csc \left(x-\frac{\pi }{3}\right)-\cot \left(x-\frac{\pi }{3}\right)|+c\)
• C. \(\frac{1}{2}\log |\csc \left(x+\frac{\pi }{3}\right)-\cot \left(x-\frac{\pi }{3}\right)|+c\)
• D. \(\frac{1}{2}\log |\csc \left(x+\frac{\pi }{3}\right)-\cot \left(x+\frac{\pi }{3}\right)|+c\)

Hint:

Write the denominator as 2 sin(x - pi/3) and use the standard cosec integral.

Answer 1

The Correct Option is B

Step 1: Use the R-sine form
With $a=1,\ b=-\sqrt3$: $R=\sqrt{1+3}=2$ and the phase shift is $-\dfrac\pi3$.

Step 2: Substitute
Let $\theta=x-\dfrac\pi3$; then $d\theta=dx$ and the integral becomes $\dfrac12\int\csc\theta\,d\theta$.

Step 3: Standard integral
$\dfrac12\log|\csc\theta-\cot\theta|+c$, which equals $\dfrac12\log|\tan\frac\theta2|+c$.

Final Answer:
The integral is $\frac12\log|\csc(x-\frac\pi3)-\cot(x-\frac\pi3)|+c$. \[ \boxed{\text{(B) }\dfrac{1}{2}\log |\csc \left(x-\dfrac{\pi }{3}\right)-\cot \left(x-\dfrac{\pi }{3}\right)|+c} \]