置換積分

$3$x=g $t$$ の関係に置換したとき，

$3$\displaystyle{ \begin{array}{lll} \int f $x$ dx & =\int f $x$ \frac{\hspace{2} dx \hspace{2}}{\hspace{2} dt \hspace{2}}dt & \vspace{6}\\ & =\int f $ g \(t$ \) {g}^{\prime } $t$ dt & \vspace{6}\\ \end{array} }$

$3$\displaystyle{ \begin{array}{lll} \int _{a}^{b} f $x$ dx & =\int _{{\alpha}}^{{\beta}} f $x$ \frac{\hspace{2} dx \hspace{2}}{\hspace{2} dt \hspace{2}}dt & \vspace{6}\\ &\displaystyle{\displaystyle{=\int _{{\alpha}}^{{\beta}} f $ g \(t$ \) {g}^{\prime } $t$ dt }}& \vspace{6}\\ \end{array} }$

ただし， $3$g ${\alpha}$ =a$ ， $3$g ${\beta}$ =b$

■計算例

$3$\displaystyle{\begin{array}{lll}\displaystyle{ \int 2t{ $ {t}^{2}+3 $ }^{2}dt }& \vspace{6}\\ \text{\hspace{5}}=\displaystyle{ \int { $ {t}^{2}+3 $ }^{2}\cdot 2tdt }& \vspace{6}\\ \end{array}}$

$3$\displaystyle{{t}^{2}+3=x}$ とおくと，

　 $3$\displaystyle{\frac{\hspace{2} dx \hspace{2}}{\hspace{2} dt \hspace{2}}=2t}$

よって，

　 $3$\displaystyle{\begin{array}{lll}=\displaystyle{ \int {x}^{2}\frac{\hspace{2} dx \hspace{2}}{\hspace{2} dt \hspace{2}}dt }& \vspace{6}\\ =\displaystyle{ \int {x}^{2}dx }& \vspace{6}\\ =\frac{\hspace{2}1\hspace{2}}{\hspace{2}3\hspace{2}}{x}^{3}+C& \vspace{6}\\ =\frac{\hspace{2}1\hspace{2}}{\hspace{2}3\hspace{2}}{ $ {t}^{2}+3 $ }^{3}+C& \vspace{6}\\ \end{array}}$

詳しくはPCサイト参照のこと。

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