$$$\cos^{4}{\left(7 x \right)}$$$ 的积分

求$$$\int \cos^{4}{\left(7 x \right)}\, dx$$$。

设$$$u=7 x$$$。

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

所以，

$${\color{red}{\int{\cos^{4}{\left(7 x \right)} d x}}} = {\color{red}{\int{\frac{\cos^{4}{\left(u \right)}}{7} d u}}}$$

对 $$$c=\frac{1}{7}$$$ 和 $$$f{\left(u \right)} = \cos^{4}{\left(u \right)}$$$ 应用常数倍法则 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$：

$${\color{red}{\int{\frac{\cos^{4}{\left(u \right)}}{7} d u}}} = {\color{red}{\left(\frac{\int{\cos^{4}{\left(u \right)} d u}}{7}\right)}}$$

应用降幂公式 $$$\cos^{4}{\left(\alpha \right)} = \frac{\cos{\left(2 \alpha \right)}}{2} + \frac{\cos{\left(4 \alpha \right)}}{8} + \frac{3}{8}$$$，并令 $$$\alpha= u $$$:

$$\frac{{\color{red}{\int{\cos^{4}{\left(u \right)} d u}}}}{7} = \frac{{\color{red}{\int{\left(\frac{\cos{\left(2 u \right)}}{2} + \frac{\cos{\left(4 u \right)}}{8} + \frac{3}{8}\right)d u}}}}{7}$$

对 $$$c=\frac{1}{8}$$$ 和 $$$f{\left(u \right)} = 4 \cos{\left(2 u \right)} + \cos{\left(4 u \right)} + 3$$$ 应用常数倍法则 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$：

$$\frac{{\color{red}{\int{\left(\frac{\cos{\left(2 u \right)}}{2} + \frac{\cos{\left(4 u \right)}}{8} + \frac{3}{8}\right)d u}}}}{7} = \frac{{\color{red}{\left(\frac{\int{\left(4 \cos{\left(2 u \right)} + \cos{\left(4 u \right)} + 3\right)d u}}{8}\right)}}}{7}$$

逐项积分：

$$\frac{{\color{red}{\int{\left(4 \cos{\left(2 u \right)} + \cos{\left(4 u \right)} + 3\right)d u}}}}{56} = \frac{{\color{red}{\left(\int{3 d u} + \int{4 \cos{\left(2 u \right)} d u} + \int{\cos{\left(4 u \right)} d u}\right)}}}{56}$$

应用常数法则 $$$\int c\, du = c u$$$，使用 $$$c=3$$$：

$$\frac{\int{4 \cos{\left(2 u \right)} d u}}{56} + \frac{\int{\cos{\left(4 u \right)} d u}}{56} + \frac{{\color{red}{\int{3 d u}}}}{56} = \frac{\int{4 \cos{\left(2 u \right)} d u}}{56} + \frac{\int{\cos{\left(4 u \right)} d u}}{56} + \frac{{\color{red}{\left(3 u\right)}}}{56}$$

对 $$$c=4$$$ 和 $$$f{\left(u \right)} = \cos{\left(2 u \right)}$$$ 应用常数倍法则 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$：

$$\frac{3 u}{56} + \frac{\int{\cos{\left(4 u \right)} d u}}{56} + \frac{{\color{red}{\int{4 \cos{\left(2 u \right)} d u}}}}{56} = \frac{3 u}{56} + \frac{\int{\cos{\left(4 u \right)} d u}}{56} + \frac{{\color{red}{\left(4 \int{\cos{\left(2 u \right)} d u}\right)}}}{56}$$

设$$$v=2 u$$$。

则$$$dv=\left(2 u\right)^{\prime }du = 2 du$$$ (步骤见»)，并有$$$du = \frac{dv}{2}$$$。

积分变为

$$\frac{3 u}{56} + \frac{\int{\cos{\left(4 u \right)} d u}}{56} + \frac{{\color{red}{\int{\cos{\left(2 u \right)} d u}}}}{14} = \frac{3 u}{56} + \frac{\int{\cos{\left(4 u \right)} d u}}{56} + \frac{{\color{red}{\int{\frac{\cos{\left(v \right)}}{2} d v}}}}{14}$$

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

$$\frac{3 u}{56} + \frac{\int{\cos{\left(4 u \right)} d u}}{56} + \frac{{\color{red}{\int{\frac{\cos{\left(v \right)}}{2} d v}}}}{14} = \frac{3 u}{56} + \frac{\int{\cos{\left(4 u \right)} d u}}{56} + \frac{{\color{red}{\left(\frac{\int{\cos{\left(v \right)} d v}}{2}\right)}}}{14}$$

余弦函数的积分为 $$$\int{\cos{\left(v \right)} d v} = \sin{\left(v \right)}$$$：

$$\frac{3 u}{56} + \frac{\int{\cos{\left(4 u \right)} d u}}{56} + \frac{{\color{red}{\int{\cos{\left(v \right)} d v}}}}{28} = \frac{3 u}{56} + \frac{\int{\cos{\left(4 u \right)} d u}}{56} + \frac{{\color{red}{\sin{\left(v \right)}}}}{28}$$

回忆一下 $$$v=2 u$$$:

$$\frac{3 u}{56} + \frac{\int{\cos{\left(4 u \right)} d u}}{56} + \frac{\sin{\left({\color{red}{v}} \right)}}{28} = \frac{3 u}{56} + \frac{\int{\cos{\left(4 u \right)} d u}}{56} + \frac{\sin{\left({\color{red}{\left(2 u\right)}} \right)}}{28}$$

设$$$v=4 u$$$。

则$$$dv=\left(4 u\right)^{\prime }du = 4 du$$$ (步骤见»)，并有$$$du = \frac{dv}{4}$$$。

该积分可以改写为

$$\frac{3 u}{56} + \frac{\sin{\left(2 u \right)}}{28} + \frac{{\color{red}{\int{\cos{\left(4 u \right)} d u}}}}{56} = \frac{3 u}{56} + \frac{\sin{\left(2 u \right)}}{28} + \frac{{\color{red}{\int{\frac{\cos{\left(v \right)}}{4} d v}}}}{56}$$

对 $$$c=\frac{1}{4}$$$ 和 $$$f{\left(v \right)} = \cos{\left(v \right)}$$$ 应用常数倍法则 $$$\int c f{\left(v \right)}\, dv = c \int f{\left(v \right)}\, dv$$$：

$$\frac{3 u}{56} + \frac{\sin{\left(2 u \right)}}{28} + \frac{{\color{red}{\int{\frac{\cos{\left(v \right)}}{4} d v}}}}{56} = \frac{3 u}{56} + \frac{\sin{\left(2 u \right)}}{28} + \frac{{\color{red}{\left(\frac{\int{\cos{\left(v \right)} d v}}{4}\right)}}}{56}$$

余弦函数的积分为 $$$\int{\cos{\left(v \right)} d v} = \sin{\left(v \right)}$$$：

$$\frac{3 u}{56} + \frac{\sin{\left(2 u \right)}}{28} + \frac{{\color{red}{\int{\cos{\left(v \right)} d v}}}}{224} = \frac{3 u}{56} + \frac{\sin{\left(2 u \right)}}{28} + \frac{{\color{red}{\sin{\left(v \right)}}}}{224}$$

回忆一下 $$$v=4 u$$$:

$$\frac{3 u}{56} + \frac{\sin{\left(2 u \right)}}{28} + \frac{\sin{\left({\color{red}{v}} \right)}}{224} = \frac{3 u}{56} + \frac{\sin{\left(2 u \right)}}{28} + \frac{\sin{\left({\color{red}{\left(4 u\right)}} \right)}}{224}$$

回忆一下 $$$u=7 x$$$:

$$\frac{\sin{\left(2 {\color{red}{u}} \right)}}{28} + \frac{\sin{\left(4 {\color{red}{u}} \right)}}{224} + \frac{3 {\color{red}{u}}}{56} = \frac{\sin{\left(2 {\color{red}{\left(7 x\right)}} \right)}}{28} + \frac{\sin{\left(4 {\color{red}{\left(7 x\right)}} \right)}}{224} + \frac{3 {\color{red}{\left(7 x\right)}}}{56}$$

因此，

$$\int{\cos^{4}{\left(7 x \right)} d x} = \frac{3 x}{8} + \frac{\sin{\left(14 x \right)}}{28} + \frac{\sin{\left(28 x \right)}}{224}$$

化简：

$$\int{\cos^{4}{\left(7 x \right)} d x} = \frac{84 x + 8 \sin{\left(14 x \right)} + \sin{\left(28 x \right)}}{224}$$

加上积分常数：

$$\int{\cos^{4}{\left(7 x \right)} d x} = \frac{84 x + 8 \sin{\left(14 x \right)} + \sin{\left(28 x \right)}}{224}+C$$

$$$\int \cos^{4}{\left(7 x \right)}\, dx = \frac{84 x + 8 \sin{\left(14 x \right)} + \sin{\left(28 x \right)}}{224} + C$$$A