부정적분과 곱 조건

함수 $f ( x )$가 이차함수이므로 $f ( x ) = ax ^{2} + bx + c$ ($a \ne 0$)라 하자. $\displaystyle g ( x ) = \int \left\{ x ^{2} + f ( x ) \right\} dx$ $\displaystyle = \int \left\{ x ^{2} + ax ^{2} + bx + c \right\} dx$ $\displaystyle = \int \left\{ ( 1 + a ) x ^{2} + bx + c \right\} dx$ $\displaystyle = \frac{1}{3} ( 1 + a ) x ^{3} + \frac{b}{2} x ^{2} + cx + C$ $\cdots \text{㉠}$ ($C$는 적분상수) 한편, $f ( x ) g ( x ) = ( ax ^{2} + bx + c ) g ( x )$ $= - 2 x ^{4} + 8 x ^{3}$ $\cdots \text{㉡}$ 이므로 $g ( x )$는 이차함수이다. ∴ $a = - 1$ ㉠, ㉡에서 $\displaystyle ( - x ^{2} + bx + c ) \left( \frac{b}{2} x ^{2} + cx + C \right)$$= - 2 x ^{4} + 8 x ^{3}$ $\displaystyle - \frac{b}{2} x ^{4} + \left( \frac{b ^{2}}{2} - c \right) x ^{3} + \left( - C + bc + \frac{bc}{2} \right) x ^{2}$ $+ \left( bC + c ^{2} ) x + c C \right.$$= - 2 x ^{4} + 8 x ^{3}$ 에서 $\displaystyle - \frac{b}{2} = - 2$, $\displaystyle \frac{b ^{2}}{2} - c = 8$, $\displaystyle - C + bc + \frac{bc}{2} = 0$, $cC = 0$ ∴ $b = 4$, $c = 0$, $C = 0$ ∴ $g ( x ) = 2 x ^{2}$ ∴ $g ( 1 ) = 2$