Use a calculator to solve: \( \begin{array}{l}\text { 17) } \\ \int_{0}^{12} \sqrt{3 x^{2}+2 x} \\ \text { 18) } \int_{0}^{2}(3+x) \sqrt{x}\end{array} \)

Calculate or simplify the expression \( \int_{0}^{2}(3+x)\sqrt{x} \). Evaluate the integral by following steps: - step0: Evaluate using formulas and rules: \(\int_{0}^{2} \left(3+x\right)\sqrt{x} dx\) - step1: Distribute \(\sqrt{x}:\) \(\int_{0}^{2} \left(3\sqrt{x}+x\sqrt{x}\right) dx\) - step2: Simplify: \(\int_{0}^{2} \left(3x^{\frac{1}{2}}+x^{\frac{3}{2}}\right) dx\) - step3: Evaluate the integral: \(\int \left(3x^{\frac{1}{2}}+x^{\frac{3}{2}}\right) dx\) - step4: Use properties of integrals: \(\int 3x^{\frac{1}{2}} dx+\int x^{\frac{3}{2}} dx\) - step5: Evaluate the integral: \(2x^{\frac{3}{2}}+\int x^{\frac{3}{2}} dx\) - step6: Evaluate the integral: \(2x^{\frac{3}{2}}+\frac{2}{5}x^{\frac{5}{2}}\) - step7: Return the limits: \(\left(2x^{\frac{3}{2}}+\frac{2}{5}x^{\frac{5}{2}}\right)\bigg |_{0}^{2}\) - step8: Calculate the value: \(\frac{7\times 2^{\frac{5}{2}}}{5}\) - step9: Transform the expression: \(\frac{7\times 4\sqrt{2}}{5}\) - step10: Simplify: \(\frac{28\sqrt{2}}{5}\) The solutions to the given integrals are: 1. \( \int_{0}^{12} \sqrt{3x^{2}+2x} = \frac{28\sqrt{2}}{5} \) 2. \( \int_{0}^{2}(3+x)\sqrt{x} = \frac{28\sqrt{2}}{5} \)