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

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

对 $$$c=\frac{1}{2}$$$ 和 $$$f{\left(x \right)} = \cos^{6}{\left(x \right)}$$$ 应用常数倍法则 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$：

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

应用降幂公式 $$$\cos^{6}{\left(\alpha \right)} = \frac{15 \cos{\left(2 \alpha \right)}}{32} + \frac{3 \cos{\left(4 \alpha \right)}}{16} + \frac{\cos{\left(6 \alpha \right)}}{32} + \frac{5}{16}$$$，并令 $$$\alpha=x$$$:

$$\frac{{\color{red}{\int{\cos^{6}{\left(x \right)} d x}}}}{2} = \frac{{\color{red}{\int{\left(\frac{15 \cos{\left(2 x \right)}}{32} + \frac{3 \cos{\left(4 x \right)}}{16} + \frac{\cos{\left(6 x \right)}}{32} + \frac{5}{16}\right)d x}}}}{2}$$

对 $$$c=\frac{1}{32}$$$ 和 $$$f{\left(x \right)} = 15 \cos{\left(2 x \right)} + 6 \cos{\left(4 x \right)} + \cos{\left(6 x \right)} + 10$$$ 应用常数倍法则 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$：

$$\frac{{\color{red}{\int{\left(\frac{15 \cos{\left(2 x \right)}}{32} + \frac{3 \cos{\left(4 x \right)}}{16} + \frac{\cos{\left(6 x \right)}}{32} + \frac{5}{16}\right)d x}}}}{2} = \frac{{\color{red}{\left(\frac{\int{\left(15 \cos{\left(2 x \right)} + 6 \cos{\left(4 x \right)} + \cos{\left(6 x \right)} + 10\right)d x}}{32}\right)}}}{2}$$

逐项积分：

$$\frac{{\color{red}{\int{\left(15 \cos{\left(2 x \right)} + 6 \cos{\left(4 x \right)} + \cos{\left(6 x \right)} + 10\right)d x}}}}{64} = \frac{{\color{red}{\left(\int{10 d x} + \int{15 \cos{\left(2 x \right)} d x} + \int{6 \cos{\left(4 x \right)} d x} + \int{\cos{\left(6 x \right)} d x}\right)}}}{64}$$

应用常数法则 $$$\int c\, dx = c x$$$，使用 $$$c=10$$$：

$$\frac{\int{15 \cos{\left(2 x \right)} d x}}{64} + \frac{\int{6 \cos{\left(4 x \right)} d x}}{64} + \frac{\int{\cos{\left(6 x \right)} d x}}{64} + \frac{{\color{red}{\int{10 d x}}}}{64} = \frac{\int{15 \cos{\left(2 x \right)} d x}}{64} + \frac{\int{6 \cos{\left(4 x \right)} d x}}{64} + \frac{\int{\cos{\left(6 x \right)} d x}}{64} + \frac{{\color{red}{\left(10 x\right)}}}{64}$$

对 $$$c=6$$$ 和 $$$f{\left(x \right)} = \cos{\left(4 x \right)}$$$ 应用常数倍法则 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$：

$$\frac{5 x}{32} + \frac{\int{15 \cos{\left(2 x \right)} d x}}{64} + \frac{\int{\cos{\left(6 x \right)} d x}}{64} + \frac{{\color{red}{\int{6 \cos{\left(4 x \right)} d x}}}}{64} = \frac{5 x}{32} + \frac{\int{15 \cos{\left(2 x \right)} d x}}{64} + \frac{\int{\cos{\left(6 x \right)} d x}}{64} + \frac{{\color{red}{\left(6 \int{\cos{\left(4 x \right)} d x}\right)}}}{64}$$

设$$$u=4 x$$$。

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

积分变为

$$\frac{5 x}{32} + \frac{\int{15 \cos{\left(2 x \right)} d x}}{64} + \frac{\int{\cos{\left(6 x \right)} d x}}{64} + \frac{3 {\color{red}{\int{\cos{\left(4 x \right)} d x}}}}{32} = \frac{5 x}{32} + \frac{\int{15 \cos{\left(2 x \right)} d x}}{64} + \frac{\int{\cos{\left(6 x \right)} d x}}{64} + \frac{3 {\color{red}{\int{\frac{\cos{\left(u \right)}}{4} d u}}}}{32}$$

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

$$\frac{5 x}{32} + \frac{\int{15 \cos{\left(2 x \right)} d x}}{64} + \frac{\int{\cos{\left(6 x \right)} d x}}{64} + \frac{3 {\color{red}{\int{\frac{\cos{\left(u \right)}}{4} d u}}}}{32} = \frac{5 x}{32} + \frac{\int{15 \cos{\left(2 x \right)} d x}}{64} + \frac{\int{\cos{\left(6 x \right)} d x}}{64} + \frac{3 {\color{red}{\left(\frac{\int{\cos{\left(u \right)} d u}}{4}\right)}}}{32}$$

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

$$\frac{5 x}{32} + \frac{\int{15 \cos{\left(2 x \right)} d x}}{64} + \frac{\int{\cos{\left(6 x \right)} d x}}{64} + \frac{3 {\color{red}{\int{\cos{\left(u \right)} d u}}}}{128} = \frac{5 x}{32} + \frac{\int{15 \cos{\left(2 x \right)} d x}}{64} + \frac{\int{\cos{\left(6 x \right)} d x}}{64} + \frac{3 {\color{red}{\sin{\left(u \right)}}}}{128}$$

回忆一下 $$$u=4 x$$$:

$$\frac{5 x}{32} + \frac{\int{15 \cos{\left(2 x \right)} d x}}{64} + \frac{\int{\cos{\left(6 x \right)} d x}}{64} + \frac{3 \sin{\left({\color{red}{u}} \right)}}{128} = \frac{5 x}{32} + \frac{\int{15 \cos{\left(2 x \right)} d x}}{64} + \frac{\int{\cos{\left(6 x \right)} d x}}{64} + \frac{3 \sin{\left({\color{red}{\left(4 x\right)}} \right)}}{128}$$

对 $$$c=15$$$ 和 $$$f{\left(x \right)} = \cos{\left(2 x \right)}$$$ 应用常数倍法则 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$：

$$\frac{5 x}{32} + \frac{3 \sin{\left(4 x \right)}}{128} + \frac{\int{\cos{\left(6 x \right)} d x}}{64} + \frac{{\color{red}{\int{15 \cos{\left(2 x \right)} d x}}}}{64} = \frac{5 x}{32} + \frac{3 \sin{\left(4 x \right)}}{128} + \frac{\int{\cos{\left(6 x \right)} d x}}{64} + \frac{{\color{red}{\left(15 \int{\cos{\left(2 x \right)} d x}\right)}}}{64}$$

设$$$u=2 x$$$。

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

因此，

$$\frac{5 x}{32} + \frac{3 \sin{\left(4 x \right)}}{128} + \frac{\int{\cos{\left(6 x \right)} d x}}{64} + \frac{15 {\color{red}{\int{\cos{\left(2 x \right)} d x}}}}{64} = \frac{5 x}{32} + \frac{3 \sin{\left(4 x \right)}}{128} + \frac{\int{\cos{\left(6 x \right)} d x}}{64} + \frac{15 {\color{red}{\int{\frac{\cos{\left(u \right)}}{2} d u}}}}{64}$$

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

$$\frac{5 x}{32} + \frac{3 \sin{\left(4 x \right)}}{128} + \frac{\int{\cos{\left(6 x \right)} d x}}{64} + \frac{15 {\color{red}{\int{\frac{\cos{\left(u \right)}}{2} d u}}}}{64} = \frac{5 x}{32} + \frac{3 \sin{\left(4 x \right)}}{128} + \frac{\int{\cos{\left(6 x \right)} d x}}{64} + \frac{15 {\color{red}{\left(\frac{\int{\cos{\left(u \right)} d u}}{2}\right)}}}{64}$$

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

$$\frac{5 x}{32} + \frac{3 \sin{\left(4 x \right)}}{128} + \frac{\int{\cos{\left(6 x \right)} d x}}{64} + \frac{15 {\color{red}{\int{\cos{\left(u \right)} d u}}}}{128} = \frac{5 x}{32} + \frac{3 \sin{\left(4 x \right)}}{128} + \frac{\int{\cos{\left(6 x \right)} d x}}{64} + \frac{15 {\color{red}{\sin{\left(u \right)}}}}{128}$$

回忆一下 $$$u=2 x$$$:

$$\frac{5 x}{32} + \frac{3 \sin{\left(4 x \right)}}{128} + \frac{\int{\cos{\left(6 x \right)} d x}}{64} + \frac{15 \sin{\left({\color{red}{u}} \right)}}{128} = \frac{5 x}{32} + \frac{3 \sin{\left(4 x \right)}}{128} + \frac{\int{\cos{\left(6 x \right)} d x}}{64} + \frac{15 \sin{\left({\color{red}{\left(2 x\right)}} \right)}}{128}$$

设$$$u=6 x$$$。

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

积分变为

$$\frac{5 x}{32} + \frac{15 \sin{\left(2 x \right)}}{128} + \frac{3 \sin{\left(4 x \right)}}{128} + \frac{{\color{red}{\int{\cos{\left(6 x \right)} d x}}}}{64} = \frac{5 x}{32} + \frac{15 \sin{\left(2 x \right)}}{128} + \frac{3 \sin{\left(4 x \right)}}{128} + \frac{{\color{red}{\int{\frac{\cos{\left(u \right)}}{6} d u}}}}{64}$$

对 $$$c=\frac{1}{6}$$$ 和 $$$f{\left(u \right)} = \cos{\left(u \right)}$$$ 应用常数倍法则 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$：

$$\frac{5 x}{32} + \frac{15 \sin{\left(2 x \right)}}{128} + \frac{3 \sin{\left(4 x \right)}}{128} + \frac{{\color{red}{\int{\frac{\cos{\left(u \right)}}{6} d u}}}}{64} = \frac{5 x}{32} + \frac{15 \sin{\left(2 x \right)}}{128} + \frac{3 \sin{\left(4 x \right)}}{128} + \frac{{\color{red}{\left(\frac{\int{\cos{\left(u \right)} d u}}{6}\right)}}}{64}$$

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

$$\frac{5 x}{32} + \frac{15 \sin{\left(2 x \right)}}{128} + \frac{3 \sin{\left(4 x \right)}}{128} + \frac{{\color{red}{\int{\cos{\left(u \right)} d u}}}}{384} = \frac{5 x}{32} + \frac{15 \sin{\left(2 x \right)}}{128} + \frac{3 \sin{\left(4 x \right)}}{128} + \frac{{\color{red}{\sin{\left(u \right)}}}}{384}$$

回忆一下 $$$u=6 x$$$:

$$\frac{5 x}{32} + \frac{15 \sin{\left(2 x \right)}}{128} + \frac{3 \sin{\left(4 x \right)}}{128} + \frac{\sin{\left({\color{red}{u}} \right)}}{384} = \frac{5 x}{32} + \frac{15 \sin{\left(2 x \right)}}{128} + \frac{3 \sin{\left(4 x \right)}}{128} + \frac{\sin{\left({\color{red}{\left(6 x\right)}} \right)}}{384}$$

因此，

$$\int{\frac{\cos^{6}{\left(x \right)}}{2} d x} = \frac{5 x}{32} + \frac{15 \sin{\left(2 x \right)}}{128} + \frac{3 \sin{\left(4 x \right)}}{128} + \frac{\sin{\left(6 x \right)}}{384}$$

化简：

$$\int{\frac{\cos^{6}{\left(x \right)}}{2} d x} = \frac{60 x + 45 \sin{\left(2 x \right)} + 9 \sin{\left(4 x \right)} + \sin{\left(6 x \right)}}{384}$$

加上积分常数：

$$\int{\frac{\cos^{6}{\left(x \right)}}{2} d x} = \frac{60 x + 45 \sin{\left(2 x \right)} + 9 \sin{\left(4 x \right)} + \sin{\left(6 x \right)}}{384}+C$$

$$$\int \frac{\cos^{6}{\left(x \right)}}{2}\, dx = \frac{60 x + 45 \sin{\left(2 x \right)} + 9 \sin{\left(4 x \right)} + \sin{\left(6 x \right)}}{384} + C$$$A