定積分

■問題

次の定積分の値を求めよ.

$3$\displaystyle{ \displaystyle{ \int _{1}^{e} x\log xdx } }$

■解説動画

◇積分の動画一覧のページへ

■答

$3$\displaystyle{\frac{\hspace{2} {e}^{2}+1 \hspace{2}}{\hspace{2}4\hspace{2}}}$

■ヒント

部分積分法を利用して解く．

今回の問題は， $3$\displaystyle{f$x$}$ と $3$\displaystyle{g$x$}$ の関係を逆にした表現の[定積分]

$3$\displaystyle{\displaystyle{ \int _{a}^{b} {f}^{\prime }$x$g$x$ }dx= \[ f$x$g$x$ \] _{a}^{b}}$ $3$\displaystyle{- \displaystyle{ \int _{a}^{b} f$x${g}^{\prime }$x$ }dx}$

を利用する.

■解説

$3$\displaystyle{\displaystyle{ \int _{1}^{e} x\log x }dx}$

$3$\displaystyle{=\displaystyle{ \int _{1}^{e} { $ \frac{\hspace{2} {x}^{2} \hspace{2}}{\hspace{2}2\hspace{2}} $ }^{\prime }\cdot \log x }dx}$

$3$\displaystyle{= \[ \frac{\hspace{2} {x}^{2} \hspace{2}}{\hspace{2}2\hspace{2}}\log x \] _{1}^{e}- \displaystyle{ \int _{1}^{e} \frac{\hspace{2}1\hspace{2}}{\hspace{2}x\hspace{2}} }\cdot \frac{\hspace{2} {x}^{2} \hspace{2}}{\hspace{2}2\hspace{2}}dx}$

$3$\displaystyle{= $ \frac{\hspace{2} {e}^{2} \hspace{2}}{\hspace{2}2\hspace{2}}\log e- \frac{\hspace{2} {1}^{2} \hspace{2}}{\hspace{2}2\hspace{2}}\log 1 $ - \displaystyle{ \int _{1}^{e} \frac{\hspace{2} x \hspace{2}}{\hspace{2}2\hspace{2}} }dx}$

$3$\displaystyle{= $ \frac{\hspace{2} {e}^{2} \hspace{2}}{\hspace{2}2\hspace{2}}\cdot 1- 0 $ - \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}\displaystyle{ \int _{1}^{e} x }dx}$

$3$\displaystyle{=\frac{\hspace{2} {e}^{2} \hspace{2}}{\hspace{2}2\hspace{2}}- \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}} \[ \frac{\hspace{2} {x}^{2} \hspace{2}}{\hspace{2}2\hspace{2}} \] _{1}^{e}}$

$3$\displaystyle{=\frac{\hspace{2} {e}^{2} \hspace{2}}{\hspace{2}2\hspace{2}}- \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}} $ \frac{\hspace{2} {e}^{2} \hspace{2}}{\hspace{2}2\hspace{2}}- \frac{\hspace{2} {1}^{2} \hspace{2}}{\hspace{2}2\hspace{2}} $ }$

$3$\displaystyle{=\frac{\hspace{2} {e}^{2} \hspace{2}}{\hspace{2}2\hspace{2}}- \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}\cdot \frac{\hspace{2} {e}^{2}- 1 \hspace{2}}{\hspace{2}2\hspace{2}}}$

$3$\displaystyle{=\frac{\hspace{2} {e}^{2} \hspace{2}}{\hspace{2}2\hspace{2}}- \frac{\hspace{2} {e}^{2}- 1 \hspace{2}}{\hspace{2}4\hspace{2}}}$

$3$\displaystyle{=\frac{\hspace{2} 2{e}^{2}- {e}^{2}+1 \hspace{2}}{\hspace{2}4\hspace{2}}}$

$3$\displaystyle{=\frac{\hspace{2} {e}^{2}+1 \hspace{2}}{\hspace{2}4\hspace{2}}}$

ホーム>>カテゴリー分類>>積分>>積分の問題>>定積分の問題>> $3$\displaystyle{ \displaystyle{ \int _{1}^{e} x\log xdx } }$

最終更新日：2025年6月9日