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

$3$\displaystyle{C}$ は積分定数

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

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

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