$$$\sin{\left(x \right)} \sec^{2}{\left(\cos{\left(x \right)} \right)}$$$ 的積分

求$$$\int \sin{\left(x \right)} \sec^{2}{\left(\cos{\left(x \right)} \right)}\, dx$$$。

令 $$$u=\cos{\left(x \right)}$$$。

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

因此，

$${\color{red}{\int{\sin{\left(x \right)} \sec^{2}{\left(\cos{\left(x \right)} \right)} d x}}} = {\color{red}{\int{\left(- \sec^{2}{\left(u \right)}\right)d u}}}$$

套用常數倍法則 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$，使用 $$$c=-1$$$ 與 $$$f{\left(u \right)} = \sec^{2}{\left(u \right)}$$$：

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

$$$\sec^{2}{\left(u \right)}$$$ 的積分是 $$$\int{\sec^{2}{\left(u \right)} d u} = \tan{\left(u \right)}$$$：

$$- {\color{red}{\int{\sec^{2}{\left(u \right)} d u}}} = - {\color{red}{\tan{\left(u \right)}}}$$

回顧一下 $$$u=\cos{\left(x \right)}$$$：

$$- \tan{\left({\color{red}{u}} \right)} = - \tan{\left({\color{red}{\cos{\left(x \right)}}} \right)}$$

因此，

$$\int{\sin{\left(x \right)} \sec^{2}{\left(\cos{\left(x \right)} \right)} d x} = - \tan{\left(\cos{\left(x \right)} \right)}$$

加上積分常數：

$$\int{\sin{\left(x \right)} \sec^{2}{\left(\cos{\left(x \right)} \right)} d x} = - \tan{\left(\cos{\left(x \right)} \right)}+C$$

$$$\int \sin{\left(x \right)} \sec^{2}{\left(\cos{\left(x \right)} \right)}\, dx = - \tan{\left(\cos{\left(x \right)} \right)} + C$$$A