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

$3$\displaystyle{C}$ は積分定数

$3$\displaystyle{{\sin }^{ - 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} \sqrt{ 1- { $ ax+b $ }^{2} } \hspace{2}}dx }& \vspace{6}\\ \text{\hspace{5}}=\frac{\hspace{2}1\hspace{2}}{\hspace{2}a\hspace{2}}{\sin }^{ - 1 } $ ax+b $ +C& \vspace{6}\\ \end{array}}$
　 $3$\displaystyle{a}$ ， $3$\displaystyle{b}$ は定数

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

■計算例
$3$\displaystyle{\begin{array}{lll}\displaystyle{ \int \frac{\hspace{2}1\hspace{2}}{\hspace{2} \sqrt{ 9- {x}^{2} } \hspace{2}}dx }& \vspace{6}\\ \text{\hspace{5}}=\displaystyle{ \int \frac{\hspace{2} \frac{\hspace{2}1\hspace{2}}{\hspace{2}3\hspace{2}}\cdot 1 \hspace{2}}{\hspace{2} \frac{\hspace{2}1\hspace{2}}{\hspace{2}3\hspace{2}}\cdot \sqrt{ 9- {x}^{2} } \hspace{2}}dx }& \vspace{6}\\ \text{\hspace{5}}=\displaystyle{ \int \frac{\hspace{2} \frac{\hspace{2}1\hspace{2}}{\hspace{2}3\hspace{2}} \hspace{2}}{\hspace{2} \sqrt{ { $ \frac{\hspace{2}1\hspace{2}}{\hspace{2}3\hspace{2}} $ }^{2} $ 9- {x}^{2} $ } \hspace{2}}dx }& \vspace{6}\\ \text{\hspace{5}}=\frac{\hspace{2}1\hspace{2}}{\hspace{2}3\hspace{2}}\displaystyle{ \int \frac{\hspace{2}1\hspace{2}}{\hspace{2} \sqrt{ 1- { $ \frac{\hspace{2}x\hspace{2}}{\hspace{2}3\hspace{2}} $ }^{2} } \hspace{2}}dx }& \vspace{6}\\ \text{\hspace{5}}=\frac{\hspace{2}1\hspace{2}}{\hspace{2}3\hspace{2}}\cdot \frac{\hspace{2}1\hspace{2}}{\hspace{2} \frac{\hspace{2}1\hspace{2}}{\hspace{2}3\hspace{2}} \hspace{2}}{\sin }^{ - 1 }\frac{\hspace{2}x\hspace{2}}{\hspace{2}3\hspace{2}}+C& \vspace{6}\\ \text{\hspace{5}}={\sin }^{ - 1 }\frac{\hspace{2}x\hspace{2}}{\hspace{2}3\hspace{2}}+C& \vspace{6}\\ \end{array}}$