Question

Evaluate the integral: \[ \int \left[\frac{1}{\cos x} - \frac{1}{\sin x} - \frac{1}{\sin x + 3\cos x}\right] dx \]

• A. \( \frac{1}{3} \log \left| \frac{\sin x}{\sin x + 3\cos x} \right| + C \)
• B. \( \log \left| \frac{\cos x}{\sin x + 3\cos x} \right| + C \)
• C. \( \frac{1}{3} \log \left| \frac{\cos x}{\sin x + 3\cos x} \right| + C \)
• D. \( \log \left| \frac{\sin x}{\sin x + 3\cos x} \right| + C \)

Hint:

For integrals involving trigonometric fractions, use substitution to simplify before integrating directly.

Answer 1

The Correct Option is C

Rewriting the integral: \[ \int \left[\frac{1}{\cos x} - \frac{1}{\sin x} - \frac{1}{\sin x + 3\cos x}\right] dx \] Using trigonometric substitutions and logarithmic properties, simplifying: \[ \frac{1}{3} \log \left| \frac{\cos x}{\sin x + 3\cos x} \right| + C \] Thus, the correct answer is: \[ \frac{1}{3} \log \left| \frac{\cos x}{\sin x + 3\cos x} \right| + C \]