1. **State the problem:** We need to evaluate the integral $$\int x(3x^2 + 5)\,dx$$. 2. **Use the distributive property:** Multiply $x$ inside the parentheses: $$x(3x^2 + 5) = 3x^3 + 5x$$ 3. **Rewrite the integral:** $$\int (3x^3 + 5x)\,dx$$ 4. **Integrate term-by-term:** - For $3x^3$, use the power rule: $$\int x^n dx = \frac{x^{n+1}}{n+1} + C$$ - For $5x$, similarly apply the power rule. 5. **Calculate each integral:** $$\int 3x^3 dx = 3 \cdot \frac{x^{4}}{4} = \frac{3}{4}x^{4}$$ $$\int 5x dx = 5 \cdot \frac{x^{2}}{2} = \frac{5}{2}x^{2}$$ 6. **Combine results and add constant of integration:** $$\int x(3x^2 + 5) dx = \frac{3}{4}x^{4} + \frac{5}{2}x^{2} + C$$ **Final answer:** $$\boxed{\frac{3}{4}x^{4} + \frac{5}{2}x^{2} + C}$$