$$$\cos^{2}{\left(8 x \right)}$$$ 的积分

求$$$\int \cos^{2}{\left(8 x \right)}\, dx$$$。

设$$$u=8 x$$$。

则$$$du=\left(8 x\right)^{\prime }dx = 8 dx$$$ (步骤见»)，并有$$$dx = \frac{du}{8}$$$。

因此，

$${\color{red}{\int{\cos^{2}{\left(8 x \right)} d x}}} = {\color{red}{\int{\frac{\cos^{2}{\left(u \right)}}{8} d u}}}$$

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

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

应用降幂公式 $$$\cos^{2}{\left(\alpha \right)} = \frac{\cos{\left(2 \alpha \right)}}{2} + \frac{1}{2}$$$，并令 $$$\alpha= u $$$:

$$\frac{{\color{red}{\int{\cos^{2}{\left(u \right)} d u}}}}{8} = \frac{{\color{red}{\int{\left(\frac{\cos{\left(2 u \right)}}{2} + \frac{1}{2}\right)d u}}}}{8}$$

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

$$\frac{{\color{red}{\int{\left(\frac{\cos{\left(2 u \right)}}{2} + \frac{1}{2}\right)d u}}}}{8} = \frac{{\color{red}{\left(\frac{\int{\left(\cos{\left(2 u \right)} + 1\right)d u}}{2}\right)}}}{8}$$

逐项积分：

$$\frac{{\color{red}{\int{\left(\cos{\left(2 u \right)} + 1\right)d u}}}}{16} = \frac{{\color{red}{\left(\int{1 d u} + \int{\cos{\left(2 u \right)} d u}\right)}}}{16}$$

应用常数法则 $$$\int c\, du = c u$$$，使用 $$$c=1$$$：

$$\frac{\int{\cos{\left(2 u \right)} d u}}{16} + \frac{{\color{red}{\int{1 d u}}}}{16} = \frac{\int{\cos{\left(2 u \right)} d u}}{16} + \frac{{\color{red}{u}}}{16}$$

设$$$v=2 u$$$。

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

所以，

$$\frac{u}{16} + \frac{{\color{red}{\int{\cos{\left(2 u \right)} d u}}}}{16} = \frac{u}{16} + \frac{{\color{red}{\int{\frac{\cos{\left(v \right)}}{2} d v}}}}{16}$$

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

$$\frac{u}{16} + \frac{{\color{red}{\int{\frac{\cos{\left(v \right)}}{2} d v}}}}{16} = \frac{u}{16} + \frac{{\color{red}{\left(\frac{\int{\cos{\left(v \right)} d v}}{2}\right)}}}{16}$$

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

$$\frac{u}{16} + \frac{{\color{red}{\int{\cos{\left(v \right)} d v}}}}{32} = \frac{u}{16} + \frac{{\color{red}{\sin{\left(v \right)}}}}{32}$$

回忆一下 $$$v=2 u$$$:

$$\frac{u}{16} + \frac{\sin{\left({\color{red}{v}} \right)}}{32} = \frac{u}{16} + \frac{\sin{\left({\color{red}{\left(2 u\right)}} \right)}}{32}$$

回忆一下 $$$u=8 x$$$:

$$\frac{\sin{\left(2 {\color{red}{u}} \right)}}{32} + \frac{{\color{red}{u}}}{16} = \frac{\sin{\left(2 {\color{red}{\left(8 x\right)}} \right)}}{32} + \frac{{\color{red}{\left(8 x\right)}}}{16}$$

因此，

$$\int{\cos^{2}{\left(8 x \right)} d x} = \frac{x}{2} + \frac{\sin{\left(16 x \right)}}{32}$$

加上积分常数：

$$\int{\cos^{2}{\left(8 x \right)} d x} = \frac{x}{2} + \frac{\sin{\left(16 x \right)}}{32}+C$$

$$$\int \cos^{2}{\left(8 x \right)}\, dx = \left(\frac{x}{2} + \frac{\sin{\left(16 x \right)}}{32}\right) + C$$$A