$$$1 - \sec^{2}{\left(x \right)}$$$ 的积分

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

逐项积分：

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

应用常数法则 $$$\int c\, dx = c x$$$，使用 $$$c=1$$$：

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

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

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

因此，

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

加上积分常数：

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

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