Answer
12
Work Step by Step
\[\begin{gathered}
\int_0^2 {\left( {3{x^2} + 2x} \right)} dx \hfill \\
\hfill \\
{\text{property }}\int {\left( {f + g} \right)dx} = \int f dx + \int {gdx} \hfill \\
\hfill \\
\int_0^2 {3{x^2}} dx + \int_0^2 {2x} dx \hfill \\
\hfill \\
3\int_0^2 {{x^2}} dx + 2\int_0^2 x dx \hfill \\
\hfill \\
{\text{Integrate using the rule }}\int {{x^n}} dx = \frac{{{x^{n + 1}}}}{{n + 1}} + C \hfill \\
\hfill \\
3\left[ {\frac{{{x^3}}}{3}} \right]_0^2 + 2\left[ {\frac{{{x^2}}}{2}} \right]_0^2 \hfill \\
\hfill \\
{\text{Use the fundamental Theorem of calculus}} \hfill \\
\hfill \\
3\left[ {\frac{{{{\left( 2 \right)}^3}}}{3} - \frac{{{{\left( 0 \right)}^3}}}{3}} \right] + 2\left[ {\frac{{{{\left( 2 \right)}^2}}}{2} - \frac{{{{\left( 0 \right)}^2}}}{3}} \right] \hfill \\
\hfill \\
{\text{Simplify}} \hfill \\
\hfill \\
3\left[ {\frac{8}{3}} \right] + 2\left[ {\frac{4}{2}} \right] \hfill \\
\hfill \\
8 + 4 \hfill \\
\hfill \\
12 \hfill \\
\end{gathered} \]
