Question - Definite Integral of a Quadratic Function

\( \int x^2 + x - 9 \, dx = \frac{1}{3}x^3 + \frac{1}{2}x^2 - 9x + C \)

\( \left[ \frac{1}{3}x^3 + \frac{1}{2}x^2 - 9x \right]_{5}^{10} \)

\( = \left( \frac{1}{3}(10)^3 + \frac{1}{2}(10)^2 - 9(10) \right) - \left( \frac{1}{3}(5)^3 + \frac{1}{2}(5)^2 - 9(5) \right) \)

\( = \left( \frac{1}{3}(1000) + \frac{1}{2}(100) - 90 \right) - \left( \frac{1}{3}(125) + \frac{1}{2}(25) - 45 \right) \)

\( = \left( 333\frac{1}{3} + 50 - 90 \right) - \left( 41\frac{2}{3} + 12\frac{1}{2} - 45 \right) \)

\( = 293\frac{1}{3} - 8\frac{5}{6} \)

\( = 293\frac{1}{3} - 8\frac{10}{12} \)

\( = 293\frac{4}{12} - 8\frac{10}{12} \)

\( = 284\frac{6}{12} \)

\( = 284\frac{1}{2} \)