部分積分

$3$\displaystyle{\begin{array}{lll}\displaystyle{ \int {f}^{\prime } $x$ g $x$ dx }& \vspace{6}\\ =f $x$ g $x$ - \displaystyle{ \int f $x$ {g}^{\prime } $x$ dx }& \vspace{6}\\ \end{array}}$

$3$\displaystyle{\begin{array}{lll}\displaystyle{ \int _{a}^{b} {f}^{\prime } $x$ g $x$ dx }& \vspace{6}\\ = \[ f $x$ g $x$ \] _{a}^{b}- \displaystyle{ \int _{a}^{b} f $x$ {g}^{\prime } $x$ dx }& \vspace{6}\\ \end{array}}$

■計算例

$3$\displaystyle{\begin{array}{lll}\displaystyle{ \int x{e}^{x}dx }& \vspace{6}\\ \text{\hspace{5}}=\displaystyle{ \int { $ {e}^{x} $ }^{\prime }xdx }& \vspace{6}\\ \end{array}}$

部分積分の公式を適用すると，

　 $3$\displaystyle{\begin{array}{lll}={e}^{x}\cdot x- \displaystyle{ \int {e}^{x}\cdot {x}^{\prime }dx }& \vspace{6}\\ =x{e}^{x}- \displaystyle{ \int {e}^{x}dx }& \vspace{6}\\ =x{e}^{x}- {e}^{x}+C& \vspace{6}\\ = $ x- 1 $ {e}^{x}+C& \vspace{6}\\ \end{array}}$

詳しくはPCサイト参照のこと。

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