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

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

改写被积函数:

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

设$$$u=\sec{\left(x \right)}$$$。

则$$$du=\left(\sec{\left(x \right)}\right)^{\prime }dx = \tan{\left(x \right)} \sec{\left(x \right)} dx$$$ (步骤见»)，并有$$$\tan{\left(x \right)} \sec{\left(x \right)} dx = du$$$。

因此，

$${\color{red}{\int{\tan{\left(x \right)} \sec{\left(x \right)} d x}}} = {\color{red}{\int{1 d u}}}$$

应用常数法则 $$$\int c\, du = c u$$$，使用 $$$c=1$$$：

$${\color{red}{\int{1 d u}}} = {\color{red}{u}}$$

回忆一下 $$$u=\sec{\left(x \right)}$$$:

$${\color{red}{u}} = {\color{red}{\sec{\left(x \right)}}}$$

因此，

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

加上积分常数：

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

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