부정적분

함수 $f \left( x \right)$의 한 부정적분이 $F \left( x \right)$이므로 $F \left( 2 \right) - F \left( 0 \right)$$\displaystyle = \int _{0} ^{2} f \left( x \right) dx$ $\therefore$ $F \left( 2 \right)$$\displaystyle = F \left( 0 \right) + \int _{0} ^{2} \left( 4 x ^{3} - 2 x \right) dx$ $\displaystyle = 4 + \left[ \begin{array}{l} \begin{matrix} \\ \end{matrix} x ^{4} - x ^{2} \end{array} \right] _{0} ^{2}$$= 4 + \left( 2 ^{4} - 2 ^{2} \right)$$= 16$ [다른 풀이] $F \left( x \right)$$\displaystyle = \int _{} ^{} {f \left( x \right) dx}$$\displaystyle = \int _{} ^{} {\left( 4 x ^{3} - 2 x \right) dx}$ $= x ^{4} - x ^{2} + C$ (단, $C$는 적분상수) $F \left( 0 \right) = 4$이므로 $0 - 0 + C = 4$, $C = 4$ $F \left( x \right)$$= x ^{4} - x ^{2} + 4$이므로 $F \left( 2 \right) = 2 ^{4} - 2 ^{2} + 4 = 16 - 4 + 4 = 16$