1. Problem: Find the indefinite integral \(\int (x^3 + \sqrt{x}) \, dx\). 2. Formula: The integral of a sum is the sum of the integrals, and \(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\) for \(n \neq -1\). 3. Work: - Rewrite \(\sqrt{x}\) as \(x^{1/2}\). - Integrate each term: \[\int x^3 \, dx = \frac{x^{4}}{4} + C_1\] \[\int x^{1/2} \, dx = \frac{x^{3/2}}{3/2} + C_2 = \frac{2}{3} x^{3/2} + C_2\] 4. Combine results: \[\int (x^3 + \sqrt{x}) \, dx = \frac{x^{4}}{4} + \frac{2}{3} x^{3/2} + C\] Final answer: \[\boxed{\frac{x^{4}}{4} + \frac{2}{3} x^{3/2} + C}\]