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\begin{aligned} &\int_{0}^{2}\left(x^{2}+2 x+1\right) \mathrm{dx}=\mathrm{F}(2)-\mathrm{F}(0)=\left\{\frac{1}{3} 2^{3}+2^{2}+2\right\}-\left\{\frac{1}{3} 0^{3}+0^{2}+0\right\} \\ &=\left\{\frac{26}{3}\right\}-\{0\}=\frac{26}{3}=8 \frac{2}{3} \end{aligned}