1. **State the problem:** We need to evaluate the integral $$\int 15x \sqrt{x} + 4 \, dx$$. 2. **Rewrite the integral:** Recall that $$\sqrt{x} = x^{\frac{1}{2}}$$, so the integral becomes: $$\int 15x \cdot x^{\frac{1}{2}} + 4 \, dx = \int 15x^{1 + \frac{1}{2}} + 4 \, dx = \int 15x^{\frac{3}{2}} + 4 \, dx$$. 3. **Split the integral:** $$\int 15x^{\frac{3}{2}} + 4 \, dx = \int 15x^{\frac{3}{2}} \, dx + \int 4 \, dx$$. 4. **Use the power rule for integration:** For $$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$$, where $$n \neq -1$$. 5. **Integrate each term:** - For $$\int 15x^{\frac{3}{2}} \, dx$$: $$15 \int x^{\frac{3}{2}} \, dx = 15 \cdot \frac{x^{\frac{3}{2} + 1}}{\frac{3}{2} + 1} = 15 \cdot \frac{x^{\frac{5}{2}}}{\frac{5}{2}}$$. - Simplify the fraction: $$15 \cdot \frac{x^{\frac{5}{2}}}{\frac{5}{2}} = 15 \cdot x^{\frac{5}{2}} \cdot \frac{2}{5} = \cancel{15} \cdot \frac{2}{\cancel{5}} x^{\frac{5}{2}} = 6x^{\frac{5}{2}}$$. - For $$\int 4 \, dx$$: $$4x$$. 6. **Combine the results and add the constant of integration:** $$\int 15x \sqrt{x} + 4 \, dx = 6x^{\frac{5}{2}} + 4x + C$$. **Final answer:** $$6x^{\frac{5}{2}} + 4x + C$$