What is int (4 x^5 -7 x^4 + 9 x^3 -8 x^2 +2 x + 1) dx?

Use the formula $\int \left({x}^{n}\right) \mathrm{dx} = \frac{{x}^{n + 1}}{n + 1} + C$.
$\implies \frac{2}{3} {x}^{6} - \frac{7}{5} {x}^{5} + \frac{9}{4} {x}^{4} - \frac{8}{3} {x}^{3} + {x}^{2} + x + C$