積分の公式

$3$\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， $3$C$ は積分定数

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

$3$\displaystyle{\frac{\hspace{2}1\hspace{2}}{\hspace{2}a\hspace{2}}{\tan }^{ - 1 }\frac{\hspace{2}x\hspace{2}}{\hspace{2}a\hspace{2}}}$ を微分すると，

$3$\displaystyle{\begin{array}{lll}{ $ \frac{\hspace{2}1\hspace{2}}{\hspace{2}a\hspace{2}}{ \tan }^{ - 1 }\frac{\hspace{2}x\hspace{2}}{\hspace{2}a\hspace{2}} $ }^{\prime }& \vspace{6}\\ =\frac{\hspace{2}1\hspace{2}}{\hspace{2}a\hspace{2}}{ $ { \tan }^{ - 1 }\frac{\hspace{2}x\hspace{2}}{\hspace{2}a\hspace{2}} $ }^{\prime }& \vspace{6}\\ =\frac{\hspace{2}1\hspace{2}}{\hspace{2}a\hspace{2}}\cdot \frac{\hspace{2}1\hspace{2}}{\hspace{2} 1+{ $ \frac{\hspace{2}x\hspace{2}}{\hspace{2}a\hspace{2}} $ }^{2} \hspace{2}}\cdot { $ \frac{\hspace{2}x\hspace{2}}{\hspace{2}a\hspace{2}} $ }^{\prime }& \vspace{6}\\ =\frac{\hspace{2}1\hspace{2}}{\hspace{2}a\hspace{2}}\cdot \frac{\hspace{2}1\hspace{2}}{\hspace{2} 1+\frac{\hspace{2} {x}^{2} \hspace{2}}{\hspace{2} {a}^{2} \hspace{2}} \hspace{2}}\cdot \frac{\hspace{2}1\hspace{2}}{\hspace{2}a\hspace{2}}& \vspace{6}\\ =\frac{\hspace{2}1\hspace{2}}{\hspace{2} {a}^{2}+{x}^{2} \hspace{2}}& \vspace{6}\\ =\frac{\hspace{2}1\hspace{2}}{\hspace{2} {x}^{2}+{a}^{2} \hspace{2}}& \vspace{6}\\ \end{array}}$

となる。よって，上の公式が成り立つ。

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【表示調整】
現在の水平表示画素数