Question

Evaluate the integral: \[ I = \int \frac{\csc x}{3\cos x + 4\sin x} dx. \]

• A. \( \frac{1}{2} \log \left| \frac{\cos x}{3\sin x + 4\cos x} \right| + C \)
• B. \( \frac{1}{3} \log \left| \frac{\sin x}{3\cos x + 4\sin x} \right| + C \)
• C. \( \frac{1}{3} \log \left| \frac{3\cos x + \sin x}{3\cos x + 4\sin x} \right| + C \)
• D. \( \frac{1}{2} \log \left| \frac{\cos x + 4\sin x}{3\cos x + 4\sin x} \right| + C \)

Hint:

For integrals of the form \( \int \frac{\csc x}{A \cos x + B \sin x} dx \), use trigonometric substitution followed by logarithmic integration.

Answer 1

The Correct Option is B

Step 1: Substituting \( u = 3\cos x + 4\sin x \) Let: \[ u = 3\cos x + 4\sin x. \] Differentiating both sides: \[ du = (-3\sin x + 4\cos x) dx. \] Rewriting the integral: \[ I = \int \frac{\csc x dx}{u}. \] Using the logarithmic integration formula, \[ \int \frac{du}{u} = \ln |u| + C, \] we obtain: \[ I = \frac{1}{3} \log \left| \frac{\sin x}{3\cos x + 4\sin x} \right| + C. \]