\(y = x(x^{4} + x + 1) = x^{5} + x^{2} + x\)

\(\int \limits_{0} ^{1} (x^{5} + x^{2} + x) \mathrm d x = \frac{x^{6}}{6} + \frac{x^{3}}{3} + \frac{x^{2}}{2}\)

= \([\frac{x^{6}}{6} + \frac{x^{3}}{3} + \frac{x^{2}}{2}]_{0} ^{1}\)

= \(\frac{1}{6} + \frac{1}{3} + \frac{1}{2}\)

= \(1\)