$$$\frac{1}{\cos^{2}{\left(x y \right)}}$$$ 关于$$$x$$$的积分

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

设$$$u=x y$$$。

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

因此，

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

对 $$$c=\frac{1}{y}$$$ 和 $$$f{\left(u \right)} = \frac{1}{\cos^{2}{\left(u \right)}}$$$ 应用常数倍法则 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$：

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

用正割表示被积函数:

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

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

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

回忆一下 $$$u=x y$$$:

$$\frac{\tan{\left({\color{red}{u}} \right)}}{y} = \frac{\tan{\left({\color{red}{x y}} \right)}}{y}$$

因此，

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

加上积分常数：

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

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