Answer
\[\frac{{{x^7}}}{7} + {x^4} + {x^3} + C\]
Work Step by Step
\[\begin{gathered}
\int_{}^{} {{x^2}\,\left( {{x^4} + 4x + 3} \right)dx} \hfill \\
Multiplying \hfill \\
\int_{}^{} {\,\left( {{x^6} + 4{x^3} + 3{x^2}} \right)dx} \hfill \\
using\,\,the\,\,sum\,\,and\,difference\,\,rules \hfill \\
\int_{}^{} {{x^6}dx} + \int_{}^{} {4{x^3}dx} + \int_{}^{} {3{x^2}dx} \hfill \\
Use\,\,the\,\,power\,\,rule \hfill \\
\int_{}^{} {{x^n}dx} = \frac{{{x^{n + 1}}}}{{n + 1}} + C \hfill \\
\frac{{{x^{6 + 1}}}}{{6 + 1}} + 4\,\left( {\frac{{{x^{3 + 1}}}}{{3 + 1}}} \right) + 3\,\left( {\frac{{{x^2} + 1}}{{2 + 1}}} \right) + C \hfill \\
\frac{{{x^7}}}{7} + {x^4} + {x^3} + C \hfill \\
\end{gathered} \]
