$$$\cos^{6}{\left(3 x \right)}$$$ 的積分

求$$$\int \cos^{6}{\left(3 x \right)}\, dx$$$。

令 $$$u=3 x$$$。

則 $$$du=\left(3 x\right)^{\prime }dx = 3 dx$$$ (步驟見»)，並可得 $$$dx = \frac{du}{3}$$$。

所以，

$${\color{red}{\int{\cos^{6}{\left(3 x \right)} d x}}} = {\color{red}{\int{\frac{\cos^{6}{\left(u \right)}}{3} d u}}}$$

套用常數倍法則 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$，使用 $$$c=\frac{1}{3}$$$ 與 $$$f{\left(u \right)} = \cos^{6}{\left(u \right)}$$$：

$${\color{red}{\int{\frac{\cos^{6}{\left(u \right)}}{3} d u}}} = {\color{red}{\left(\frac{\int{\cos^{6}{\left(u \right)} d u}}{3}\right)}}$$

套用降冪公式 $$$\cos^{6}{\left(\alpha \right)} = \frac{15 \cos{\left(2 \alpha \right)}}{32} + \frac{3 \cos{\left(4 \alpha \right)}}{16} + \frac{\cos{\left(6 \alpha \right)}}{32} + \frac{5}{16}$$$，令 $$$\alpha= u $$$:

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

套用常數倍法則 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$，使用 $$$c=\frac{1}{32}$$$ 與 $$$f{\left(u \right)} = 15 \cos{\left(2 u \right)} + 6 \cos{\left(4 u \right)} + \cos{\left(6 u \right)} + 10$$$：

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

逐項積分：

$$\frac{{\color{red}{\int{\left(15 \cos{\left(2 u \right)} + 6 \cos{\left(4 u \right)} + \cos{\left(6 u \right)} + 10\right)d u}}}}{96} = \frac{{\color{red}{\left(\int{10 d u} + \int{15 \cos{\left(2 u \right)} d u} + \int{6 \cos{\left(4 u \right)} d u} + \int{\cos{\left(6 u \right)} d u}\right)}}}{96}$$

配合 $$$c=10$$$，應用常數法則 $$$\int c\, du = c u$$$：

$$\frac{\int{15 \cos{\left(2 u \right)} d u}}{96} + \frac{\int{6 \cos{\left(4 u \right)} d u}}{96} + \frac{\int{\cos{\left(6 u \right)} d u}}{96} + \frac{{\color{red}{\int{10 d u}}}}{96} = \frac{\int{15 \cos{\left(2 u \right)} d u}}{96} + \frac{\int{6 \cos{\left(4 u \right)} d u}}{96} + \frac{\int{\cos{\left(6 u \right)} d u}}{96} + \frac{{\color{red}{\left(10 u\right)}}}{96}$$

套用常數倍法則 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$，使用 $$$c=6$$$ 與 $$$f{\left(u \right)} = \cos{\left(4 u \right)}$$$：

$$\frac{5 u}{48} + \frac{\int{15 \cos{\left(2 u \right)} d u}}{96} + \frac{\int{\cos{\left(6 u \right)} d u}}{96} + \frac{{\color{red}{\int{6 \cos{\left(4 u \right)} d u}}}}{96} = \frac{5 u}{48} + \frac{\int{15 \cos{\left(2 u \right)} d u}}{96} + \frac{\int{\cos{\left(6 u \right)} d u}}{96} + \frac{{\color{red}{\left(6 \int{\cos{\left(4 u \right)} d u}\right)}}}{96}$$

令 $$$v=4 u$$$。

則 $$$dv=\left(4 u\right)^{\prime }du = 4 du$$$ (步驟見»)，並可得 $$$du = \frac{dv}{4}$$$。

因此，

$$\frac{5 u}{48} + \frac{\int{15 \cos{\left(2 u \right)} d u}}{96} + \frac{\int{\cos{\left(6 u \right)} d u}}{96} + \frac{{\color{red}{\int{\cos{\left(4 u \right)} d u}}}}{16} = \frac{5 u}{48} + \frac{\int{15 \cos{\left(2 u \right)} d u}}{96} + \frac{\int{\cos{\left(6 u \right)} d u}}{96} + \frac{{\color{red}{\int{\frac{\cos{\left(v \right)}}{4} d v}}}}{16}$$

套用常數倍法則 $$$\int c f{\left(v \right)}\, dv = c \int f{\left(v \right)}\, dv$$$，使用 $$$c=\frac{1}{4}$$$ 與 $$$f{\left(v \right)} = \cos{\left(v \right)}$$$：

$$\frac{5 u}{48} + \frac{\int{15 \cos{\left(2 u \right)} d u}}{96} + \frac{\int{\cos{\left(6 u \right)} d u}}{96} + \frac{{\color{red}{\int{\frac{\cos{\left(v \right)}}{4} d v}}}}{16} = \frac{5 u}{48} + \frac{\int{15 \cos{\left(2 u \right)} d u}}{96} + \frac{\int{\cos{\left(6 u \right)} d u}}{96} + \frac{{\color{red}{\left(\frac{\int{\cos{\left(v \right)} d v}}{4}\right)}}}{16}$$

餘弦函數的積分為 $$$\int{\cos{\left(v \right)} d v} = \sin{\left(v \right)}$$$：

$$\frac{5 u}{48} + \frac{\int{15 \cos{\left(2 u \right)} d u}}{96} + \frac{\int{\cos{\left(6 u \right)} d u}}{96} + \frac{{\color{red}{\int{\cos{\left(v \right)} d v}}}}{64} = \frac{5 u}{48} + \frac{\int{15 \cos{\left(2 u \right)} d u}}{96} + \frac{\int{\cos{\left(6 u \right)} d u}}{96} + \frac{{\color{red}{\sin{\left(v \right)}}}}{64}$$

回顧一下 $$$v=4 u$$$：

$$\frac{5 u}{48} + \frac{\int{15 \cos{\left(2 u \right)} d u}}{96} + \frac{\int{\cos{\left(6 u \right)} d u}}{96} + \frac{\sin{\left({\color{red}{v}} \right)}}{64} = \frac{5 u}{48} + \frac{\int{15 \cos{\left(2 u \right)} d u}}{96} + \frac{\int{\cos{\left(6 u \right)} d u}}{96} + \frac{\sin{\left({\color{red}{\left(4 u\right)}} \right)}}{64}$$

套用常數倍法則 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$，使用 $$$c=15$$$ 與 $$$f{\left(u \right)} = \cos{\left(2 u \right)}$$$：

$$\frac{5 u}{48} + \frac{\sin{\left(4 u \right)}}{64} + \frac{\int{\cos{\left(6 u \right)} d u}}{96} + \frac{{\color{red}{\int{15 \cos{\left(2 u \right)} d u}}}}{96} = \frac{5 u}{48} + \frac{\sin{\left(4 u \right)}}{64} + \frac{\int{\cos{\left(6 u \right)} d u}}{96} + \frac{{\color{red}{\left(15 \int{\cos{\left(2 u \right)} d u}\right)}}}{96}$$

令 $$$v=2 u$$$。

則 $$$dv=\left(2 u\right)^{\prime }du = 2 du$$$ (步驟見»)，並可得 $$$du = \frac{dv}{2}$$$。

該積分可改寫為

$$\frac{5 u}{48} + \frac{\sin{\left(4 u \right)}}{64} + \frac{\int{\cos{\left(6 u \right)} d u}}{96} + \frac{5 {\color{red}{\int{\cos{\left(2 u \right)} d u}}}}{32} = \frac{5 u}{48} + \frac{\sin{\left(4 u \right)}}{64} + \frac{\int{\cos{\left(6 u \right)} d u}}{96} + \frac{5 {\color{red}{\int{\frac{\cos{\left(v \right)}}{2} d v}}}}{32}$$

套用常數倍法則 $$$\int c f{\left(v \right)}\, dv = c \int f{\left(v \right)}\, dv$$$，使用 $$$c=\frac{1}{2}$$$ 與 $$$f{\left(v \right)} = \cos{\left(v \right)}$$$：

$$\frac{5 u}{48} + \frac{\sin{\left(4 u \right)}}{64} + \frac{\int{\cos{\left(6 u \right)} d u}}{96} + \frac{5 {\color{red}{\int{\frac{\cos{\left(v \right)}}{2} d v}}}}{32} = \frac{5 u}{48} + \frac{\sin{\left(4 u \right)}}{64} + \frac{\int{\cos{\left(6 u \right)} d u}}{96} + \frac{5 {\color{red}{\left(\frac{\int{\cos{\left(v \right)} d v}}{2}\right)}}}{32}$$

餘弦函數的積分為 $$$\int{\cos{\left(v \right)} d v} = \sin{\left(v \right)}$$$：

$$\frac{5 u}{48} + \frac{\sin{\left(4 u \right)}}{64} + \frac{\int{\cos{\left(6 u \right)} d u}}{96} + \frac{5 {\color{red}{\int{\cos{\left(v \right)} d v}}}}{64} = \frac{5 u}{48} + \frac{\sin{\left(4 u \right)}}{64} + \frac{\int{\cos{\left(6 u \right)} d u}}{96} + \frac{5 {\color{red}{\sin{\left(v \right)}}}}{64}$$

回顧一下 $$$v=2 u$$$：

$$\frac{5 u}{48} + \frac{\sin{\left(4 u \right)}}{64} + \frac{\int{\cos{\left(6 u \right)} d u}}{96} + \frac{5 \sin{\left({\color{red}{v}} \right)}}{64} = \frac{5 u}{48} + \frac{\sin{\left(4 u \right)}}{64} + \frac{\int{\cos{\left(6 u \right)} d u}}{96} + \frac{5 \sin{\left({\color{red}{\left(2 u\right)}} \right)}}{64}$$

令 $$$v=6 u$$$。

則 $$$dv=\left(6 u\right)^{\prime }du = 6 du$$$ (步驟見»)，並可得 $$$du = \frac{dv}{6}$$$。

因此，

$$\frac{5 u}{48} + \frac{5 \sin{\left(2 u \right)}}{64} + \frac{\sin{\left(4 u \right)}}{64} + \frac{{\color{red}{\int{\cos{\left(6 u \right)} d u}}}}{96} = \frac{5 u}{48} + \frac{5 \sin{\left(2 u \right)}}{64} + \frac{\sin{\left(4 u \right)}}{64} + \frac{{\color{red}{\int{\frac{\cos{\left(v \right)}}{6} d v}}}}{96}$$

套用常數倍法則 $$$\int c f{\left(v \right)}\, dv = c \int f{\left(v \right)}\, dv$$$，使用 $$$c=\frac{1}{6}$$$ 與 $$$f{\left(v \right)} = \cos{\left(v \right)}$$$：

$$\frac{5 u}{48} + \frac{5 \sin{\left(2 u \right)}}{64} + \frac{\sin{\left(4 u \right)}}{64} + \frac{{\color{red}{\int{\frac{\cos{\left(v \right)}}{6} d v}}}}{96} = \frac{5 u}{48} + \frac{5 \sin{\left(2 u \right)}}{64} + \frac{\sin{\left(4 u \right)}}{64} + \frac{{\color{red}{\left(\frac{\int{\cos{\left(v \right)} d v}}{6}\right)}}}{96}$$

餘弦函數的積分為 $$$\int{\cos{\left(v \right)} d v} = \sin{\left(v \right)}$$$：

$$\frac{5 u}{48} + \frac{5 \sin{\left(2 u \right)}}{64} + \frac{\sin{\left(4 u \right)}}{64} + \frac{{\color{red}{\int{\cos{\left(v \right)} d v}}}}{576} = \frac{5 u}{48} + \frac{5 \sin{\left(2 u \right)}}{64} + \frac{\sin{\left(4 u \right)}}{64} + \frac{{\color{red}{\sin{\left(v \right)}}}}{576}$$

回顧一下 $$$v=6 u$$$：

$$\frac{5 u}{48} + \frac{5 \sin{\left(2 u \right)}}{64} + \frac{\sin{\left(4 u \right)}}{64} + \frac{\sin{\left({\color{red}{v}} \right)}}{576} = \frac{5 u}{48} + \frac{5 \sin{\left(2 u \right)}}{64} + \frac{\sin{\left(4 u \right)}}{64} + \frac{\sin{\left({\color{red}{\left(6 u\right)}} \right)}}{576}$$

回顧一下 $$$u=3 x$$$：

$$\frac{5 \sin{\left(2 {\color{red}{u}} \right)}}{64} + \frac{\sin{\left(4 {\color{red}{u}} \right)}}{64} + \frac{\sin{\left(6 {\color{red}{u}} \right)}}{576} + \frac{5 {\color{red}{u}}}{48} = \frac{5 \sin{\left(2 {\color{red}{\left(3 x\right)}} \right)}}{64} + \frac{\sin{\left(4 {\color{red}{\left(3 x\right)}} \right)}}{64} + \frac{\sin{\left(6 {\color{red}{\left(3 x\right)}} \right)}}{576} + \frac{5 {\color{red}{\left(3 x\right)}}}{48}$$

因此，

$$\int{\cos^{6}{\left(3 x \right)} d x} = \frac{5 x}{16} + \frac{5 \sin{\left(6 x \right)}}{64} + \frac{\sin{\left(12 x \right)}}{64} + \frac{\sin{\left(18 x \right)}}{576}$$

化簡：

$$\int{\cos^{6}{\left(3 x \right)} d x} = \frac{180 x + 45 \sin{\left(6 x \right)} + 9 \sin{\left(12 x \right)} + \sin{\left(18 x \right)}}{576}$$

加上積分常數：

$$\int{\cos^{6}{\left(3 x \right)} d x} = \frac{180 x + 45 \sin{\left(6 x \right)} + 9 \sin{\left(12 x \right)} + \sin{\left(18 x \right)}}{576}+C$$

$$$\int \cos^{6}{\left(3 x \right)}\, dx = \frac{180 x + 45 \sin{\left(6 x \right)} + 9 \sin{\left(12 x \right)} + \sin{\left(18 x \right)}}{576} + C$$$A