Answer
\[ = \frac{{199}}{{187}}\]
Work Step by Step
\[\begin{gathered}
\int_0^1 {\,\left( {4{x^{21}} - 2{x^{16}} + 1} \right)\,dx} \hfill \\
\hfill \\
integrate \hfill \\
\hfill \\
\,\,\left[ {\frac{{2{x^{22}}}}{{11}} - \frac{{2{x^{17}}}}{{17}} + x} \right]_0^1 \hfill \\
\hfill \\
Fundamental\,\,theorem \hfill \\
\hfill \\
= \frac{{2\,{{\left( 1 \right)}^{22}}}}{{11}} - \frac{{2\,{{\left( 1 \right)}^{17}}}}{{17}} + 1 - 0 \hfill \\
\hfill \\
= \frac{2}{{11}} - \frac{2}{{17}} + 1 \hfill \\
\hfill \\
simplify \hfill \\
\hfill \\
= \frac{{199}}{{187}} \hfill \\
\end{gathered} \]