$$$\sec^{2}{\left(\frac{x}{6} \right)}$$$ 的积分

求$$$\int \sec^{2}{\left(\frac{x}{6} \right)}\, dx$$$。

设$$$u=\frac{x}{6}$$$。

则$$$du=\left(\frac{x}{6}\right)^{\prime }dx = \frac{dx}{6}$$$ (步骤见»)，并有$$$dx = 6 du$$$。

积分变为

$${\color{red}{\int{\sec^{2}{\left(\frac{x}{6} \right)} d x}}} = {\color{red}{\int{6 \sec^{2}{\left(u \right)} d u}}}$$

对 $$$c=6$$$ 和 $$$f{\left(u \right)} = \sec^{2}{\left(u \right)}$$$ 应用常数倍法则 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$：

$${\color{red}{\int{6 \sec^{2}{\left(u \right)} d u}}} = {\color{red}{\left(6 \int{\sec^{2}{\left(u \right)} d u}\right)}}$$

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

$$6 {\color{red}{\int{\sec^{2}{\left(u \right)} d u}}} = 6 {\color{red}{\tan{\left(u \right)}}}$$

回忆一下 $$$u=\frac{x}{6}$$$:

$$6 \tan{\left({\color{red}{u}} \right)} = 6 \tan{\left({\color{red}{\left(\frac{x}{6}\right)}} \right)}$$

因此，

$$\int{\sec^{2}{\left(\frac{x}{6} \right)} d x} = 6 \tan{\left(\frac{x}{6} \right)}$$

加上积分常数：

$$\int{\sec^{2}{\left(\frac{x}{6} \right)} d x} = 6 \tan{\left(\frac{x}{6} \right)}+C$$

$$$\int \sec^{2}{\left(\frac{x}{6} \right)}\, dx = 6 \tan{\left(\frac{x}{6} \right)} + C$$$A