적분 관계식과 함숫값

$\displaystyle \int _{1} ^{x} f \left( t \right) dt = xf \left( x \right) - x ^{3}$의 양변을 $x$에 대하여 미분하면 $f \left( x \right) = f \left( x \right) + xf' \left( x \right) - 3 x ^{2}$ $xf' \left( x \right) = 3 x ^{2}$ $f' \left( x \right) = 3 x$ $\displaystyle f \left( x \right) = \frac{3}{2} x ^{2} + C \left( \right.$단, $C$는 적분상수$\left. \right)$ $\displaystyle \int _{1} ^{x} f \left( t \right) dt = xf \left( x \right) - x ^{3}$ $x = 1$일 때, $0 = 1 \times f \left( 1 \right) - 1$, $f \left( 1 \right) = 1$ $\displaystyle f \left( 1 \right) = \frac{3}{2} + C = 1$이므로 $\displaystyle C = - \frac{1}{2}$ $\displaystyle f \left( x \right) = \frac{3}{2} x ^{2} - \frac{1}{2}$ 따라서 $\displaystyle f \left( 2 \right) = 6 - \frac{1}{2} = \frac{11}{2}$