$3$\displaystyle{\displaystyle{ \int \frac{\hspace{2}1\hspace{2}}{\hspace{2} 1+{x}^{2} \hspace{2}}dx }={\tan }^{ - 1 }x+C}$

　 $3$C$ は積分定数

$3$\displaystyle{{\tan }^{ - 1 }x}$ の微分を参照のこと。

■応用公式
$3$\displaystyle{\begin{array}{lll}\displaystyle{ \int \frac{\hspace{2}1\hspace{2}}{\hspace{2} 1+{ $ ax+b $ }^{2} \hspace{2}}dx }& \vspace{6}\\ \text{\hspace{5}}=- \frac{\hspace{2}1\hspace{2}}{\hspace{2}a\hspace{2}}{\tan }^{ - 1 } $ ax+b $ +C& \vspace{6}\\ \end{array}}$
　 $3$a$ ， $3$b$ は定数

$3$\displaystyle{\displaystyle{ \int \frac{\hspace{2}1\hspace{2}}{\hspace{2} {x}^{2}+{a}^{2} \hspace{2}}dx }=\frac{\hspace{2}1\hspace{2}}{\hspace{2}a\hspace{2}}{\tan }^{ - 1 }\frac{\hspace{2}x\hspace{2}}{\hspace{2}a\hspace{2}}+C}$
　 $3$a$ は定数⇒

■計算例
$3$\displaystyle{\begin{array}{lll}\displaystyle{ \int \frac{\hspace{2}1\hspace{2}}{\hspace{2} 4+{x}^{2} \hspace{2}}dx }& \vspace{6}\\ \text{\hspace{5}}=\displaystyle{ \int \frac{\hspace{2} \frac{\hspace{2}1\hspace{2}}{\hspace{2}4\hspace{2}}\cdot 1 \hspace{2}}{\hspace{2} \frac{\hspace{2}1\hspace{2}}{\hspace{2}4\hspace{2}}\cdot $ 4+{x}^{2} $ \hspace{2}}dx }& \vspace{6}\\ \text{\hspace{5}}=\displaystyle{ \int \frac{\hspace{2} \frac{\hspace{2}1\hspace{2}}{\hspace{2}4\hspace{2}} \hspace{2}}{\hspace{2} 1+{ $ \frac{\hspace{2}x\hspace{2}}{\hspace{2}2\hspace{2}} $ }^{2} \hspace{2}}dx }& \vspace{6}\\ \text{\hspace{5}}=\frac{\hspace{2}1\hspace{2}}{\hspace{2}4\hspace{2}}\displaystyle{ \int \frac{\hspace{2}1\hspace{2}}{\hspace{2} 1+{ $ \frac{\hspace{2}x\hspace{2}}{\hspace{2}2\hspace{2}} $ }^{2} \hspace{2}}dx }& \vspace{6}\\ \text{\hspace{5}}=\frac{\hspace{2}1\hspace{2}}{\hspace{2}4\hspace{2}}\cdot \frac{\hspace{2}1\hspace{2}}{\hspace{2} \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}} \hspace{2}}{\tan }^{ - 1 }\frac{\hspace{2}x\hspace{2}}{\hspace{2}2\hspace{2}}+C& \vspace{6}\\ \text{\hspace{5}}=\frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}{\tan }^{ - 1 }\frac{\hspace{2}x\hspace{2}}{\hspace{2}2\hspace{2}}+C& \vspace{6}\\ \end{array}}$