$$$\tan{\left(7 x \right)} \sec^{5}{\left(7 x \right)}$$$ 的积分

求$$$\int \tan{\left(7 x \right)} \sec^{5}{\left(7 x \right)}\, dx$$$。

设$$$u=7 x$$$。

则$$$du=\left(7 x\right)^{\prime }dx = 7 dx$$$ (步骤见»)，并有$$$dx = \frac{du}{7}$$$。

积分变为

$${\color{red}{\int{\tan{\left(7 x \right)} \sec^{5}{\left(7 x \right)} d x}}} = {\color{red}{\int{\frac{\tan{\left(u \right)} \sec^{5}{\left(u \right)}}{7} d u}}}$$

对 $$$c=\frac{1}{7}$$$ 和 $$$f{\left(u \right)} = \tan{\left(u \right)} \sec^{5}{\left(u \right)}$$$ 应用常数倍法则 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$：

$${\color{red}{\int{\frac{\tan{\left(u \right)} \sec^{5}{\left(u \right)}}{7} d u}}} = {\color{red}{\left(\frac{\int{\tan{\left(u \right)} \sec^{5}{\left(u \right)} d u}}{7}\right)}}$$

设$$$v=\sec{\left(u \right)}$$$。

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

因此，

$$\frac{{\color{red}{\int{\tan{\left(u \right)} \sec^{5}{\left(u \right)} d u}}}}{7} = \frac{{\color{red}{\int{v^{4} d v}}}}{7}$$

应用幂法则 $$$\int v^{n}\, dv = \frac{v^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$，其中 $$$n=4$$$：

$$\frac{{\color{red}{\int{v^{4} d v}}}}{7}=\frac{{\color{red}{\frac{v^{1 + 4}}{1 + 4}}}}{7}=\frac{{\color{red}{\left(\frac{v^{5}}{5}\right)}}}{7}$$

回忆一下 $$$v=\sec{\left(u \right)}$$$:

$$\frac{{\color{red}{v}}^{5}}{35} = \frac{{\color{red}{\sec{\left(u \right)}}}^{5}}{35}$$

回忆一下 $$$u=7 x$$$:

$$\frac{\sec^{5}{\left({\color{red}{u}} \right)}}{35} = \frac{\sec^{5}{\left({\color{red}{\left(7 x\right)}} \right)}}{35}$$

因此，

$$\int{\tan{\left(7 x \right)} \sec^{5}{\left(7 x \right)} d x} = \frac{\sec^{5}{\left(7 x \right)}}{35}$$

加上积分常数：

$$\int{\tan{\left(7 x \right)} \sec^{5}{\left(7 x \right)} d x} = \frac{\sec^{5}{\left(7 x \right)}}{35}+C$$

$$$\int \tan{\left(7 x \right)} \sec^{5}{\left(7 x \right)}\, dx = \frac{\sec^{5}{\left(7 x \right)}}{35} + C$$$A