Question

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

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

Hint:

\(\int \csc x dx = \ln |\tan(x/2)| + C\).

Answer 1

The Correct Option is D

To solve the integral \(\int \frac{1}{\sin\left(x - \frac{\pi}{3}\right)} \, dx\), we need to employ a trigonometric identity to simplify the integrand.

The function \(\frac{1}{\sin\left(x - \frac{\pi}{3}\right)}\) can be rewritten using the cosecant function: \(\csc\left(x - \frac{\pi}{3}\right)\).

Recall the identity for cosecant:

• \(\csc \theta = \frac{1}{\sin \theta}\)

Now, let's consider the identity for cosecant and its antiderivative:

• The integral of \(\csc u\) is:
  • \(\int \csc u \, du = -\log \left| \csc u + \cot u \right| + C\)

Using substitution, set \(u = x - \frac{\pi}{3}\) which implies \(du = dx\).

The integral becomes:

• \(\int \csc u \, du = \int \csc\left(x - \frac{\pi}{3}\right) \, dx\)

Using the known antiderivative result:

• \(=-\log \left| \csc\left(x - \frac{\pi}{3}\right) + \cot\left(x - \frac{\pi}{3}\right) \right| + C\)

This integral does not match any of the given options, confirming that the correct answer is "None of the above."