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置換積分法の例

■代表的な置換

$3$a> 0$ とする．

$3$\displaystyle{ \int \sqrt{ {a}^{2}+{x}^{2} }dx }$ $3$\Rightarrow$ $3$\displaystyle{x=a\tan {\theta}}$ 　 $3$\displaystyle{\left({ - \frac{\hspace{2}{\pi}\hspace{2}}{\hspace{2}2\hspace{2}}< {\theta}< \frac{\hspace{2}{\pi}\hspace{2}}{\hspace{2}2\hspace{2}} }\right)}$ ⇒ここを参照
$3$\displaystyle{ \int \sqrt{ {a}^{2}-{x}^{2} }dx }$ $3$\Rightarrow$ $3$x=a\sin {\theta}$ $3$\displaystyle{ \( - \frac{\hspace{2}{\pi}\hspace{2}}{\hspace{2}2\hspace{2}}\leq {\theta}\leq \frac{\hspace{2}{\pi}\hspace{2}}{\hspace{2}2\hspace{2}} \) }$ ，または $3$x=a\cos {\theta}$ $3$ \( 0\leq {\theta}\leq {\pi} \)$ ⇒ ここを参照
$3$\displaystyle{ \int \sqrt{ {x}^{2}-{a}^{2} }dx }$ $3$\Rightarrow$ $3$\displaystyle{ x=\frac{\hspace{2}a\hspace{2}}{\hspace{2} \cos {\theta} \hspace{2}} }$ 　 $3$\displaystyle{ \left({ 0< {\theta}< \frac{\hspace{2}{\pi}\hspace{2}}{\hspace{2}2\hspace{2}},\frac{\hspace{2}{\pi}\hspace{2}}{\hspace{2}2\hspace{2}}< {\theta}< {\pi} }\right) }$ ⇒ ここを参照
$3$\displaystyle{\int \frac{\hspace{2}1\hspace{2}}{\hspace{2} \sqrt{ {x}^{2}+{a}^{2} } \hspace{2}}dx}$ $3$\displaystyle{\Rightarrow \{ \begin{array}{lll} x+\sqrt{ {x}^{2}+{a}^{2} }=t & \vspace{6}\\ x=a\tan {\theta}\text{\hspace{5}}\text{\hspace{5}}\text{\hspace{5}} \( - \frac{\hspace{2}{\pi}\hspace{2}}{\hspace{2}2\hspace{2}}< {\theta}< \frac{\hspace{2}{\pi}\hspace{2}}{\hspace{2}2\hspace{2}} \) & \vspace{6}\\ \end{array} }$ 　⇒ここを参照
$3$\displaystyle{\int \frac{\hspace{2}1\hspace{2}}{\hspace{2} \sqrt{ {x}^{2}- {a}^{2} } \hspace{2}}dx}$ $3$\displaystyle{\Rightarrow \{ \begin{array}{lll} x+\sqrt{ {x}^{2}- {a}^{2} }=t & \vspace{6}\\ x=\frac{\hspace{2}a\hspace{2}}{\hspace{2} \cos {\theta} \hspace{2}}\text{\hspace{5}}\text{\hspace{5}}\text{\hspace{5}} \( 0< {\theta}< \frac{\hspace{2}{\pi}\hspace{2}}{\hspace{2}2\hspace{2}},\frac{\hspace{2}{\pi}\hspace{2}}{\hspace{2}2\hspace{2}}< {\theta}< {\pi} \) & \vspace{6}\\ \end{array} }$ 　⇒ここを参照
$3$\displaystyle{\displaystyle{ \int \frac{\hspace{2}1\hspace{2}}{\hspace{2} {a}^{2}+{x}^{2} \hspace{2}}dx }}$ $3$\displaystyle{\Rightarrow x=a\tan {\theta}\text{\hspace{5}}\left({ - \frac{\hspace{2}{\pi}\hspace{2}}{\hspace{2}2\hspace{2}}< {\theta}< \frac{\hspace{2}{\pi}\hspace{2}}{\hspace{2}2\hspace{2}} }\right)}$ ⇒ここを参照

■三角関数

$3$\displaystyle{\int f \( \sin x \) \cos xdx }$ の場合， $3$\displaystyle{\sin x=t}$ とおく． $3$\displaystyle{\cos xdx=dt}$ ⇒ここを参照
$3$\displaystyle{\int f \( \cos x \) \sin xdx }$ の場合， $3$\displaystyle{\cos x=t}$ とおく． $3$\displaystyle{\cos xdx=- dt}$ ⇒ここを参照
$3$\displaystyle{\int f \( \tan x \) \cdot \frac{\hspace{2}1\hspace{2}}{\hspace{2} { \cos }^{2}x \hspace{2}}dx }$ の場合， $3$\displaystyle{\tan x=t}$ とおく． $3$\displaystyle{\frac{\hspace{2} dt \hspace{2}}{\hspace{2} dx \hspace{2}}=\frac{\hspace{2}1\hspace{2}}{\hspace{2} { \cos }^{2}x \hspace{2}}\text{\hspace{5}}\rightarrow \text{\hspace{5}}\frac{\hspace{2}1\hspace{2}}{\hspace{2} { \cos }^{2}x \hspace{2}}dx=dt}$ ⇒ここを参照
$3$\displaystyle{\int f \( \sin x,\cos x \) dx }$ の場合， $3$\displaystyle{\tan \frac{\hspace{2}x\hspace{2}}{\hspace{2}2\hspace{2}}=t}$ とおく．
$3$\displaystyle{\sin x=\frac{\hspace{2} 2t \hspace{2}}{\hspace{2} 1+{t}^{2} \hspace{2}}}$ ， $3$\displaystyle{\cos x=\frac{\hspace{2} 1- {t}^{2} \hspace{2}}{\hspace{2} 1+{t}^{2} \hspace{2}}}$ ， $3$\displaystyle{ dx=\frac{\hspace{2}2\hspace{2}}{\hspace{2} 1+{t}^{2} \hspace{2}}dt }$ ⇒ここを参照

■置換積分法を用いた公式

$3$\displaystyle{\int \frac{\hspace{2} {f}^{\prime } \(x\) \hspace{2}}{\hspace{2} f \(x\) \hspace{2}}dx =\log \| f \(x\) \| +C}$ ⇒ここを参照
$3$\displaystyle{\int {f}^{\prime } \( g \(x\) \) {g}^{\prime } \(x\) dx =f \( g \(x\) \) +C}$ ⇒ここを参照
$3$\displaystyle{\int { \{ f \(x\) \} }^{{\alpha}}{f}^{\prime } \(x\) dx =\frac{\hspace{2} { \{ f \(x\) \} }^{ {\alpha}+1 } \hspace{2}}{\hspace{2} {\alpha}+1 \hspace{2}}+C}$ ⇒ここを参照

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最終更新日： 2025年10月6日