Question

Evaluate the integral: $$ \int \sec \left(x - \frac{\pi}{3}\right) \sec \left(x + \frac{\pi}{6}\right) dx $$

• A. \[ \log \left| \frac{\sec \left(x - \frac{\pi}{3}\right)}{\sec \left(x + \frac{\pi}{6}\right)} \right| + c \]
• B. \[ \log \left| \frac{\cos \left(x - \frac{\pi}{3}\right)}{\cos \left(x + \frac{\pi}{6}\right)} \right| + c \]
• C. \[ \log \left| \frac{\csc \left(x - \frac{\pi}{3}\right)}{\csc \left(x + \frac{\pi}{6}\right)} \right| + c \]
• D. \[ \log \left| \frac{\sin \left(x - \frac{\pi}{3}\right)}{\sin \left(x + \frac{\pi}{6}\right)} \right| + c \]

Hint:

Express secants as reciprocals of cosine, then use logarithmic differentiation.

Answer 1

The Correct Option is B

Recall that \(\sec \theta = \frac{1}{\cos \theta}\). Rewrite the integral: \[ \int \sec \left(x - \frac{\pi}{3}\right) \sec \left(x + \frac{\pi}{6}\right) dx = \int \frac{1}{\cos \left(x - \frac{\pi}{3}\right)} \cdot \frac{1}{\cos \left(x + \frac{\pi}{6}\right)} dx \] Multiply numerator and denominator appropriately and notice derivative relations. The integration results in a logarithmic function involving cosine terms: \[ = \log \left| \frac{\cos \left(x - \frac{\pi}{3}\right)}{\cos \left(x + \frac{\pi}{6}\right)} \right| + c \]