$3$\displaystyle{\displaystyle{ \int _{\alpha }^{{\beta}} $ x- \alpha $ $ x- {\beta} $ dx }}$
$3$\displaystyle{=- \frac{\hspace{2}1\hspace{2}}{\hspace{2}6\hspace{2}}{ $ {\beta}- \alpha $ }^{3}}$

$3$\displaystyle{\begin{array}{lll}\int _{{\alpha}}^{{\beta}} $ x- {\alpha} $ $ x- {\beta} $ dx& \vspace{6}\\ =\int _{{\alpha}}^{{\beta}} $ x- {\alpha} $ \{ $ x- {\alpha} $ - $ {\beta}- {\alpha} $ \} dx & \vspace{6}\\ \end{array}}$
$3$ x- {\alpha}=t$ とき置換積分をする。
$3$\displaystyle{ \frac{\hspace{2} dt \hspace{2}}{\hspace{2} dx \hspace{2}}=1 }$ より $3$\displaystyle{ dx=dt \hspace{10} \hspace{10} \hspace{10} \hspace{10} \hspace{10} \hspace{10} }$
したがって，
$3$\displaystyle{ =\int _{{\alpha}}^{{\beta}} t \{ t- $ {\beta}- {\alpha} $ \} dt }$
$3$ =\int _{{\alpha}}^{{\beta}} \{ {t}^{2}\text{-}t $ {\beta}- {\alpha} $ \} dt$
$3$\displaystyle{ = \[ \frac{\hspace{2}1\hspace{2}}{\hspace{2}3\hspace{2}}{t}^{3}- \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}{t}^{2} $ {\beta}- {\alpha} $ \] _{{\alpha}}^{{\beta}} }$
$3$\displaystyle{ = \[ \frac{\hspace{2}1\hspace{2}}{\hspace{2}3\hspace{2}}{ $ x- {\alpha} $ }^{3} }$
　 $3$\displaystyle{ \hspace{5}\hspace{5}- \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}{ $ x- {\alpha} $ }^{2} $ {\beta}- {\alpha} $ \] _{{\alpha}}^{{\beta}} }$
$3$\displaystyle{ =\frac{\hspace{2}{ $ {\beta}- {\alpha} $ }^{3}\hspace{2}}{\hspace{2}3\hspace{2}}- \frac{\hspace{2}{ $ {\beta}- {\alpha} $ }^{3}\hspace{2}}{\hspace{2}2\hspace{2}} }$
$3$\displaystyle{ =- \frac{\hspace{2}1\hspace{2}}{\hspace{2}6\hspace{2}}{ $ {\beta}- {\alpha} $ }^{\text{3}} }$