대칭구간 정적분

$\displaystyle \int _{-3} ^{3} \left( x + 1 \right) f \left( x \right) dx$$\displaystyle = 36 + \int _{-3} ^{3} f \left( x \right) dx$에서 $\displaystyle \int _{-3} ^{3} \left( x + 1 \right) f \left( x \right) dx$$\displaystyle = \int _{-3} ^{3} \left\{ x f \left( x \right) + f \left( x \right) \right\} dx$ $\displaystyle = \int _{-3} ^{3} x f \left( x \right) dx + \int _{-3} ^{3} f \left( x \right) dx$ 이므로 $\displaystyle \int _{-3} ^{3} x f \left( x \right) dx + \int _{-3} ^{3} f \left( x \right) dx$$\displaystyle = 36 + \int _{-3} ^{3} f \left( x \right) dx$ $\therefore$ $\displaystyle \int _{-3} ^{3} xf \left( x \right) dx$$= 36$ 이때 $f \left( x \right) = x ^{2} + ax$이므로 $\displaystyle \int _{-3} ^{3} xf \left( x \right) dx$$\displaystyle = \int _{-3} ^{3} \left( x ^{3} + ax ^{2} \right) dx$ $\displaystyle = 2 \int _{0} ^{3} ax ^{2} dx$ $\displaystyle = 2 \left[ \begin{array}{l} \begin{matrix} \\ \end{matrix} \frac{1}{3} ax ^{3} \end{array} \right] _{0} ^{3}$ $= 18 a$ $18 a = 36$이므로 $a = 2$