$$$\sec^{2}{\left(x y \right)}$$$ 关于$$$x$$$的积分

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

设$$$u=x y$$$。

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

积分变为

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

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

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

$$$\sec^{2}{\left(u \right)}$$$ 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，

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

加上积分常数：

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

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