原始関数を展開して
与式 $3$\displaystyle{=\int _{{\alpha}}^{{\beta}} \{ {x}^{2}- ${\alpha}+{\beta}$x+{\alpha}{\beta} \} dx }$
　　　 $3$\displaystyle{= \[ \frac{\hspace{2}1\hspace{2}}{\hspace{2}3\hspace{2}}{x}^{3}- \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}} $ {\alpha}+{\beta} $ {x}^{2}+{\alpha}{\beta}x \] _{{\alpha}}^{{\beta}}}$
　 $3$\displaystyle{=- \frac{\hspace{2}1\hspace{2}}{\hspace{2}6\hspace{2}}{{\beta}}^{3}- \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}{\alpha}{{\beta}}^{2}- $ \frac{\hspace{2}1\hspace{2}}{\hspace{2}6\hspace{2}}{{\alpha}}^{3}+\frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}{{\alpha}}^{2}{\beta} $ }$
　 $3$\displaystyle{=- \frac{\hspace{2}1\hspace{2}}{\hspace{2}6\hspace{2}}{{\beta}}^{3}- \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}{{\alpha}}^{2}{\beta}+\frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}{\alpha}{{\beta}}^{2}+\frac{\hspace{2}1\hspace{2}}{\hspace{2}6\hspace{2}}{{\alpha}}^{3}}$
　 $3$\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2}6\hspace{2}} \{ - $ {{\beta}}^{3}- 3{\alpha}{{\beta}}^{2}+3{{\alpha}}^{2}{\beta}- {{\alpha}}^{3} $ \} }$
　 $3$\displaystyle{=- \frac{\hspace{2}1\hspace{2}}{\hspace{2}6\hspace{2}}{ $ {\beta}- {\alpha} $ }^{3}}$

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