$$$\frac{\sec{\left(x \right)}}{\cos{\left(x \right)}}$$$ 的积分

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

改写被积函数:

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

$$$\sec^{2}{\left(x \right)}$$$ 的积分为 $$$\int{\sec^{2}{\left(x \right)} d x} = \tan{\left(x \right)}$$$:

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

因此，

$$\int{\frac{\sec{\left(x \right)}}{\cos{\left(x \right)}} d x} = \tan{\left(x \right)}$$

加上积分常数：

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

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