$$$\frac{1}{\sin{\left(x \right)} \tan{\left(x \right)}}$$$ 的積分

求$$$\int \frac{1}{\sin{\left(x \right)} \tan{\left(x \right)}}\, dx$$$。

令 $$$u=\sin{\left(x \right)}$$$。

則 $$$du=\left(\sin{\left(x \right)}\right)^{\prime }dx = \cos{\left(x \right)} dx$$$ (步驟見»)，並可得 $$$\cos{\left(x \right)} dx = du$$$。

因此，

$${\color{red}{\int{\frac{1}{\sin{\left(x \right)} \tan{\left(x \right)}} d x}}} = {\color{red}{\int{\frac{1}{u^{2}} d u}}}$$

套用冪次法則 $$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$，以 $$$n=-2$$$：

$${\color{red}{\int{\frac{1}{u^{2}} d u}}}={\color{red}{\int{u^{-2} d u}}}={\color{red}{\frac{u^{-2 + 1}}{-2 + 1}}}={\color{red}{\left(- u^{-1}\right)}}={\color{red}{\left(- \frac{1}{u}\right)}}$$

回顧一下 $$$u=\sin{\left(x \right)}$$$：

$$- {\color{red}{u}}^{-1} = - {\color{red}{\sin{\left(x \right)}}}^{-1}$$

因此，

$$\int{\frac{1}{\sin{\left(x \right)} \tan{\left(x \right)}} d x} = - \frac{1}{\sin{\left(x \right)}}$$

加上積分常數：

$$\int{\frac{1}{\sin{\left(x \right)} \tan{\left(x \right)}} d x} = - \frac{1}{\sin{\left(x \right)}}+C$$

$$$\int \frac{1}{\sin{\left(x \right)} \tan{\left(x \right)}}\, dx = - \frac{1}{\sin{\left(x \right)}} + C$$$A