$$$\frac{1}{\cos^{2}{\left(x \right)}}$$$ 的積分

求$$$\int \frac{1}{\cos^{2}{\left(x \right)}}\, dx$$$。

將被積函數以正割表示:

$${\color{red}{\int{\frac{1}{\cos^{2}{\left(x \right)}} d x}}} = {\color{red}{\int{\sec^{2}{\left(x \right)} d x}}}$$

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

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

因此，

$$\int{\frac{1}{\cos^{2}{\left(x \right)}} d x} = \tan{\left(x \right)}$$

加上積分常數：

$$\int{\frac{1}{\cos^{2}{\left(x \right)}} d x} = \tan{\left(x \right)}+C$$

$$$\int \frac{1}{\cos^{2}{\left(x \right)}}\, dx = \tan{\left(x \right)} + C$$$A