$$$\sin{\left(x \right)} \sin{\left(2 x \right)} \cos{\left(x \right)}$$$ 的積分

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

使用公式 $$$\sin\left(\alpha \right)\sin\left(\beta \right)=\frac{1}{2} \cos\left(\alpha-\beta \right)-\frac{1}{2} \cos\left(\alpha+\beta \right)$$$，以 $$$\alpha=x$$$ 和 $$$\beta=2 x$$$ 將 $$$\sin\left(x \right)\sin\left(2 x \right)$$$ 改寫:

$${\color{red}{\int{\sin{\left(x \right)} \sin{\left(2 x \right)} \cos{\left(x \right)} d x}}} = {\color{red}{\int{\left(\frac{\cos{\left(x \right)}}{2} - \frac{\cos{\left(3 x \right)}}{2}\right) \cos{\left(x \right)} d x}}}$$

展開該表達式:

$${\color{red}{\int{\left(\frac{\cos{\left(x \right)}}{2} - \frac{\cos{\left(3 x \right)}}{2}\right) \cos{\left(x \right)} d x}}} = {\color{red}{\int{\left(\frac{\cos^{2}{\left(x \right)}}{2} - \frac{\cos{\left(x \right)} \cos{\left(3 x \right)}}{2}\right)d x}}}$$

套用常數倍法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$，使用 $$$c=\frac{1}{2}$$$ 與 $$$f{\left(x \right)} = \cos^{2}{\left(x \right)} - \cos{\left(x \right)} \cos{\left(3 x \right)}$$$：

$${\color{red}{\int{\left(\frac{\cos^{2}{\left(x \right)}}{2} - \frac{\cos{\left(x \right)} \cos{\left(3 x \right)}}{2}\right)d x}}} = {\color{red}{\left(\frac{\int{\left(\cos^{2}{\left(x \right)} - \cos{\left(x \right)} \cos{\left(3 x \right)}\right)d x}}{2}\right)}}$$

逐項積分：

$$\frac{{\color{red}{\int{\left(\cos^{2}{\left(x \right)} - \cos{\left(x \right)} \cos{\left(3 x \right)}\right)d x}}}}{2} = \frac{{\color{red}{\left(- \int{\cos{\left(x \right)} \cos{\left(3 x \right)} d x} + \int{\cos^{2}{\left(x \right)} d x}\right)}}}{2}$$

套用降冪公式 $$$\cos^{2}{\left(\alpha \right)} = \frac{\cos{\left(2 \alpha \right)}}{2} + \frac{1}{2}$$$，令 $$$\alpha=x$$$:

$$- \frac{\int{\cos{\left(x \right)} \cos{\left(3 x \right)} d x}}{2} + \frac{{\color{red}{\int{\cos^{2}{\left(x \right)} d x}}}}{2} = - \frac{\int{\cos{\left(x \right)} \cos{\left(3 x \right)} d x}}{2} + \frac{{\color{red}{\int{\left(\frac{\cos{\left(2 x \right)}}{2} + \frac{1}{2}\right)d x}}}}{2}$$

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

$$- \frac{\int{\cos{\left(x \right)} \cos{\left(3 x \right)} d x}}{2} + \frac{{\color{red}{\int{\left(\frac{\cos{\left(2 x \right)}}{2} + \frac{1}{2}\right)d x}}}}{2} = - \frac{\int{\cos{\left(x \right)} \cos{\left(3 x \right)} d x}}{2} + \frac{{\color{red}{\left(\frac{\int{\left(\cos{\left(2 x \right)} + 1\right)d x}}{2}\right)}}}{2}$$

逐項積分：

$$- \frac{\int{\cos{\left(x \right)} \cos{\left(3 x \right)} d x}}{2} + \frac{{\color{red}{\int{\left(\cos{\left(2 x \right)} + 1\right)d x}}}}{4} = - \frac{\int{\cos{\left(x \right)} \cos{\left(3 x \right)} d x}}{2} + \frac{{\color{red}{\left(\int{1 d x} + \int{\cos{\left(2 x \right)} d x}\right)}}}{4}$$

配合 $$$c=1$$$，應用常數法則 $$$\int c\, dx = c x$$$：

$$- \frac{\int{\cos{\left(x \right)} \cos{\left(3 x \right)} d x}}{2} + \frac{\int{\cos{\left(2 x \right)} d x}}{4} + \frac{{\color{red}{\int{1 d x}}}}{4} = - \frac{\int{\cos{\left(x \right)} \cos{\left(3 x \right)} d x}}{2} + \frac{\int{\cos{\left(2 x \right)} d x}}{4} + \frac{{\color{red}{x}}}{4}$$

令 $$$u=2 x$$$。

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

該積分變為

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

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

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

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

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

回顧一下 $$$u=2 x$$$：

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

使用公式 $$$\cos\left(\alpha \right)\cos\left(\beta \right)=\frac{1}{2} \cos\left(\alpha-\beta \right)+\frac{1}{2} \cos\left(\alpha+\beta \right)$$$，令 $$$\alpha=x$$$ 與 $$$\beta=3 x$$$，將被積函數改寫:

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

套用常數倍法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$，使用 $$$c=\frac{1}{2}$$$ 與 $$$f{\left(x \right)} = \cos{\left(2 x \right)} + \cos{\left(4 x \right)}$$$：

$$\frac{x}{4} + \frac{\sin{\left(2 x \right)}}{8} - \frac{{\color{red}{\int{\left(\frac{\cos{\left(2 x \right)}}{2} + \frac{\cos{\left(4 x \right)}}{2}\right)d x}}}}{2} = \frac{x}{4} + \frac{\sin{\left(2 x \right)}}{8} - \frac{{\color{red}{\left(\frac{\int{\left(\cos{\left(2 x \right)} + \cos{\left(4 x \right)}\right)d x}}{2}\right)}}}{2}$$

逐項積分：

$$\frac{x}{4} + \frac{\sin{\left(2 x \right)}}{8} - \frac{{\color{red}{\int{\left(\cos{\left(2 x \right)} + \cos{\left(4 x \right)}\right)d x}}}}{4} = \frac{x}{4} + \frac{\sin{\left(2 x \right)}}{8} - \frac{{\color{red}{\left(\int{\cos{\left(2 x \right)} d x} + \int{\cos{\left(4 x \right)} d x}\right)}}}{4}$$

積分 $$$\int{\cos{\left(2 x \right)} d x}$$$ 已經計算過：

$$\int{\cos{\left(2 x \right)} d x} = \frac{\sin{\left(2 x \right)}}{2}$$

因此，

$$\frac{x}{4} + \frac{\sin{\left(2 x \right)}}{8} - \frac{\int{\cos{\left(4 x \right)} d x}}{4} - \frac{{\color{red}{\int{\cos{\left(2 x \right)} d x}}}}{4} = \frac{x}{4} + \frac{\sin{\left(2 x \right)}}{8} - \frac{\int{\cos{\left(4 x \right)} d x}}{4} - \frac{{\color{red}{\left(\frac{\sin{\left(2 x \right)}}{2}\right)}}}{4}$$

令 $$$v=4 x$$$。

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

因此，

$$\frac{x}{4} - \frac{{\color{red}{\int{\cos{\left(4 x \right)} d x}}}}{4} = \frac{x}{4} - \frac{{\color{red}{\int{\frac{\cos{\left(v \right)}}{4} d v}}}}{4}$$

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 $$$f{\left(v \right)} = \cos{\left(v \right)}$$$：

$$\frac{x}{4} - \frac{{\color{red}{\int{\frac{\cos{\left(v \right)}}{4} d v}}}}{4} = \frac{x}{4} - \frac{{\color{red}{\left(\frac{\int{\cos{\left(v \right)} d v}}{4}\right)}}}{4}$$

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

$$\frac{x}{4} - \frac{{\color{red}{\int{\cos{\left(v \right)} d v}}}}{16} = \frac{x}{4} - \frac{{\color{red}{\sin{\left(v \right)}}}}{16}$$

回顧一下 $$$v=4 x$$$：

$$\frac{x}{4} - \frac{\sin{\left({\color{red}{v}} \right)}}{16} = \frac{x}{4} - \frac{\sin{\left({\color{red}{\left(4 x\right)}} \right)}}{16}$$

因此，

$$\int{\sin{\left(x \right)} \sin{\left(2 x \right)} \cos{\left(x \right)} d x} = \frac{x}{4} - \frac{\sin{\left(4 x \right)}}{16}$$

加上積分常數：

$$\int{\sin{\left(x \right)} \sin{\left(2 x \right)} \cos{\left(x \right)} d x} = \frac{x}{4} - \frac{\sin{\left(4 x \right)}}{16}+C$$

$$$\int \sin{\left(x \right)} \sin{\left(2 x \right)} \cos{\left(x \right)}\, dx = \left(\frac{x}{4} - \frac{\sin{\left(4 x \right)}}{16}\right) + C$$$A