1. Problem: Calculate the integral $$\int \frac{\cos 5x}{3 + \sin 5x} \, dx$$. 2. Formula and rules: Use substitution for integrals involving trigonometric functions. Let $$u = 3 + \sin 5x$$, then $$du = 5 \cos 5x \, dx$$. 3. Intermediate work: - Rewrite the integral as $$\int \frac{\cos 5x}{3 + \sin 5x} \, dx = \int \frac{1}{u} \cdot \frac{du}{5} = \frac{1}{5} \int \frac{1}{u} \, du$$. 4. Evaluate the integral: - $$\frac{1}{5} \int \frac{1}{u} \, du = \frac{1}{5} \ln |u| + C = \frac{1}{5} \ln |3 + \sin 5x| + C$$. 5. Explanation: We used substitution to simplify the integral into a basic logarithmic form. The derivative of the denominator's inner function appears in the numerator, allowing direct substitution. Final answer: $$\int \frac{\cos 5x}{3 + \sin 5x} \, dx = \frac{1}{5} \ln |3 + \sin 5x| + C$$.