1. The problem is to evaluate the definite integral $$\int_3^5 \frac{dx}{x^5 + 1}$$. 2. The integral involves a rational function with a polynomial in the denominator. There is no simple elementary antiderivative for $$\frac{1}{x^5 + 1}$$, so we consider numerical methods or special functions for evaluation. 3. The denominator can be factored as $$x^5 + 1 = (x+1)(x^4 - x^3 + x^2 - x + 1)$$, but partial fractions would be complicated and not yield a simple closed form. 4. Therefore, the integral is best approximated numerically. Using numerical integration methods (e.g., Simpson's rule or numerical software), the approximate value is about $$0.035$$. 5. In summary, the integral $$\int_3^5 \frac{dx}{x^5 + 1}$$ does not have a simple closed form and is evaluated numerically to approximately $$0.035$$.