[tex]\displaystyle
\int_0^2 x^3 + 2x^2 -7x+1 dx= \\ \\
= \frac{x^4}{4} + \frac{2x^3}{3} - \frac{7x^2}{2}+x \left|_{_0}^{^2} \right = \\ \\
= \frac{2^4}{4} + \frac{2\cdot 2^3}{3} - \frac{7 \cdot 2^2}{2}+2 = \\ \\
= \frac{16}{4} + \frac{2\cdot 8}{3} - \frac{7 \cdot 4}{2}+2 = \\ \\
= 4+ \frac{16}{3} -7\cdot 2 + 2 = \frac{16}{3} +4-14+2 = \\ \\
=\frac{16}{3} -8= \frac{16}{3} - \frac{24}{3} =\frac{16-24}{3} = \boxed{-\frac{8}{3}}[/tex]
