정적분의 성질

$\displaystyle f \left( 0 \right) - f \left( - 1 \right) = \int _{- 1} ^{0} f' \left( x \right) dx$이므로 $\displaystyle f \left( 0 \right) - f \left( - 1 \right) + \int _{0} ^{1} \left\{ x ^{2} + 2 x + f' \left( x \right) \right\} dx$ $\displaystyle = \int _{- 1} ^{0} f' \left( x \right) dx + \int _{0} ^{1} \left( x ^{2} + 2 x \right) dx + \int _{0} ^{1} f' \left( x \right) dx$ $\displaystyle = \int _{- 1} ^{1} f' \left( x \right) dx + \int _{0} ^{1} \left( x ^{2} + 2 x \right) dx$ $\displaystyle = 0 + \left[ \begin{array}{l} \begin{matrix} \\ \end{matrix} \frac{1}{3} x ^{3} + x ^{2} \end{array} \right] _{0} ^{1}$ $\displaystyle = \left( \frac{1}{3} + 1 \right) - 0 = \frac{4}{3}$