$$$\sin{\left(2 x \right)} \sin{\left(5 x \right)} \cos{\left(2 x \right)} \cos{\left(5 x \right)}$$$ 的積分

求$$$\int \sin{\left(2 x \right)} \sin{\left(5 x \right)} \cos{\left(2 x \right)} \cos{\left(5 x \right)}\, dx$$$。

使用公式 $$$\cos\left(\alpha \right)\cos\left(\beta \right)=\frac{1}{2} \cos\left(\alpha-\beta \right)+\frac{1}{2} \cos\left(\alpha+\beta \right)$$$，以 $$$\alpha=2 x$$$ 和 $$$\beta=5 x$$$ 將 $$$\cos\left(2 x \right)\cos\left(5 x \right)$$$ 改寫:

$${\color{red}{\int{\sin{\left(2 x \right)} \sin{\left(5 x \right)} \cos{\left(2 x \right)} \cos{\left(5 x \right)} d x}}} = {\color{red}{\int{\left(\frac{\cos{\left(3 x \right)}}{2} + \frac{\cos{\left(7 x \right)}}{2}\right) \sin{\left(2 x \right)} \sin{\left(5 x \right)} d x}}}$$

展開該表達式:

$${\color{red}{\int{\left(\frac{\cos{\left(3 x \right)}}{2} + \frac{\cos{\left(7 x \right)}}{2}\right) \sin{\left(2 x \right)} \sin{\left(5 x \right)} d x}}} = {\color{red}{\int{\left(\frac{\sin{\left(2 x \right)} \sin{\left(5 x \right)} \cos{\left(3 x \right)}}{2} + \frac{\sin{\left(2 x \right)} \sin{\left(5 x \right)} \cos{\left(7 x \right)}}{2}\right)d x}}}$$

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 $$$f{\left(x \right)} = \sin{\left(2 x \right)} \sin{\left(5 x \right)} \cos{\left(3 x \right)} + \sin{\left(2 x \right)} \sin{\left(5 x \right)} \cos{\left(7 x \right)}$$$：

$${\color{red}{\int{\left(\frac{\sin{\left(2 x \right)} \sin{\left(5 x \right)} \cos{\left(3 x \right)}}{2} + \frac{\sin{\left(2 x \right)} \sin{\left(5 x \right)} \cos{\left(7 x \right)}}{2}\right)d x}}} = {\color{red}{\left(\frac{\int{\left(\sin{\left(2 x \right)} \sin{\left(5 x \right)} \cos{\left(3 x \right)} + \sin{\left(2 x \right)} \sin{\left(5 x \right)} \cos{\left(7 x \right)}\right)d x}}{2}\right)}}$$

逐項積分：

$$\frac{{\color{red}{\int{\left(\sin{\left(2 x \right)} \sin{\left(5 x \right)} \cos{\left(3 x \right)} + \sin{\left(2 x \right)} \sin{\left(5 x \right)} \cos{\left(7 x \right)}\right)d x}}}}{2} = \frac{{\color{red}{\left(\int{\sin{\left(2 x \right)} \sin{\left(5 x \right)} \cos{\left(3 x \right)} d x} + \int{\sin{\left(2 x \right)} \sin{\left(5 x \right)} \cos{\left(7 x \right)} d x}\right)}}}{2}$$

使用公式 $$$\sin\left(\alpha \right)\cos\left(\beta \right)=\frac{1}{2} \sin\left(\alpha-\beta \right)+\frac{1}{2} \sin\left(\alpha+\beta \right)$$$，以 $$$\alpha=2 x$$$ 和 $$$\beta=3 x$$$ 將 $$$\sin\left(2 x \right)\cos\left(3 x \right)$$$ 改寫:

$$\frac{\int{\sin{\left(2 x \right)} \sin{\left(5 x \right)} \cos{\left(7 x \right)} d x}}{2} + \frac{{\color{red}{\int{\sin{\left(2 x \right)} \sin{\left(5 x \right)} \cos{\left(3 x \right)} d x}}}}{2} = \frac{\int{\sin{\left(2 x \right)} \sin{\left(5 x \right)} \cos{\left(7 x \right)} d x}}{2} + \frac{{\color{red}{\int{\left(- \frac{\sin{\left(x \right)}}{2} + \frac{\sin{\left(5 x \right)}}{2}\right) \sin{\left(5 x \right)} d x}}}}{2}$$

展開該表達式:

$$\frac{\int{\sin{\left(2 x \right)} \sin{\left(5 x \right)} \cos{\left(7 x \right)} d x}}{2} + \frac{{\color{red}{\int{\left(- \frac{\sin{\left(x \right)}}{2} + \frac{\sin{\left(5 x \right)}}{2}\right) \sin{\left(5 x \right)} d x}}}}{2} = \frac{\int{\sin{\left(2 x \right)} \sin{\left(5 x \right)} \cos{\left(7 x \right)} d x}}{2} + \frac{{\color{red}{\int{\left(- \frac{\sin{\left(x \right)} \sin{\left(5 x \right)}}{2} + \frac{\sin^{2}{\left(5 x \right)}}{2}\right)d x}}}}{2}$$

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 $$$f{\left(x \right)} = - \sin{\left(x \right)} \sin{\left(5 x \right)} + \sin^{2}{\left(5 x \right)}$$$：

$$\frac{\int{\sin{\left(2 x \right)} \sin{\left(5 x \right)} \cos{\left(7 x \right)} d x}}{2} + \frac{{\color{red}{\int{\left(- \frac{\sin{\left(x \right)} \sin{\left(5 x \right)}}{2} + \frac{\sin^{2}{\left(5 x \right)}}{2}\right)d x}}}}{2} = \frac{\int{\sin{\left(2 x \right)} \sin{\left(5 x \right)} \cos{\left(7 x \right)} d x}}{2} + \frac{{\color{red}{\left(\frac{\int{\left(- \sin{\left(x \right)} \sin{\left(5 x \right)} + \sin^{2}{\left(5 x \right)}\right)d x}}{2}\right)}}}{2}$$

逐項積分：

$$\frac{\int{\sin{\left(2 x \right)} \sin{\left(5 x \right)} \cos{\left(7 x \right)} d x}}{2} + \frac{{\color{red}{\int{\left(- \sin{\left(x \right)} \sin{\left(5 x \right)} + \sin^{2}{\left(5 x \right)}\right)d x}}}}{4} = \frac{\int{\sin{\left(2 x \right)} \sin{\left(5 x \right)} \cos{\left(7 x \right)} d x}}{2} + \frac{{\color{red}{\left(- \int{\sin{\left(x \right)} \sin{\left(5 x \right)} d x} + \int{\sin^{2}{\left(5 x \right)} d x}\right)}}}{4}$$

令 $$$u=5 x$$$。

則 $$$du=\left(5 x\right)^{\prime }dx = 5 dx$$$ (步驟見»)，並可得 $$$dx = \frac{du}{5}$$$。

該積分變為

$$- \frac{\int{\sin{\left(x \right)} \sin{\left(5 x \right)} d x}}{4} + \frac{\int{\sin{\left(2 x \right)} \sin{\left(5 x \right)} \cos{\left(7 x \right)} d x}}{2} + \frac{{\color{red}{\int{\sin^{2}{\left(5 x \right)} d x}}}}{4} = - \frac{\int{\sin{\left(x \right)} \sin{\left(5 x \right)} d x}}{4} + \frac{\int{\sin{\left(2 x \right)} \sin{\left(5 x \right)} \cos{\left(7 x \right)} d x}}{2} + \frac{{\color{red}{\int{\frac{\sin^{2}{\left(u \right)}}{5} d u}}}}{4}$$

套用常數倍法則 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$，使用 $$$c=\frac{1}{5}$$$ 與 $$$f{\left(u \right)} = \sin^{2}{\left(u \right)}$$$：

$$- \frac{\int{\sin{\left(x \right)} \sin{\left(5 x \right)} d x}}{4} + \frac{\int{\sin{\left(2 x \right)} \sin{\left(5 x \right)} \cos{\left(7 x \right)} d x}}{2} + \frac{{\color{red}{\int{\frac{\sin^{2}{\left(u \right)}}{5} d u}}}}{4} = - \frac{\int{\sin{\left(x \right)} \sin{\left(5 x \right)} d x}}{4} + \frac{\int{\sin{\left(2 x \right)} \sin{\left(5 x \right)} \cos{\left(7 x \right)} d x}}{2} + \frac{{\color{red}{\left(\frac{\int{\sin^{2}{\left(u \right)} d u}}{5}\right)}}}{4}$$

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

$$- \frac{\int{\sin{\left(x \right)} \sin{\left(5 x \right)} d x}}{4} + \frac{\int{\sin{\left(2 x \right)} \sin{\left(5 x \right)} \cos{\left(7 x \right)} d x}}{2} + \frac{{\color{red}{\int{\sin^{2}{\left(u \right)} d u}}}}{20} = - \frac{\int{\sin{\left(x \right)} \sin{\left(5 x \right)} d x}}{4} + \frac{\int{\sin{\left(2 x \right)} \sin{\left(5 x \right)} \cos{\left(7 x \right)} d x}}{2} + \frac{{\color{red}{\int{\left(\frac{1}{2} - \frac{\cos{\left(2 u \right)}}{2}\right)d u}}}}{20}$$

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 $$$f{\left(u \right)} = 1 - \cos{\left(2 u \right)}$$$：

$$- \frac{\int{\sin{\left(x \right)} \sin{\left(5 x \right)} d x}}{4} + \frac{\int{\sin{\left(2 x \right)} \sin{\left(5 x \right)} \cos{\left(7 x \right)} d x}}{2} + \frac{{\color{red}{\int{\left(\frac{1}{2} - \frac{\cos{\left(2 u \right)}}{2}\right)d u}}}}{20} = - \frac{\int{\sin{\left(x \right)} \sin{\left(5 x \right)} d x}}{4} + \frac{\int{\sin{\left(2 x \right)} \sin{\left(5 x \right)} \cos{\left(7 x \right)} d x}}{2} + \frac{{\color{red}{\left(\frac{\int{\left(1 - \cos{\left(2 u \right)}\right)d u}}{2}\right)}}}{20}$$

逐項積分：

$$- \frac{\int{\sin{\left(x \right)} \sin{\left(5 x \right)} d x}}{4} + \frac{\int{\sin{\left(2 x \right)} \sin{\left(5 x \right)} \cos{\left(7 x \right)} d x}}{2} + \frac{{\color{red}{\int{\left(1 - \cos{\left(2 u \right)}\right)d u}}}}{40} = - \frac{\int{\sin{\left(x \right)} \sin{\left(5 x \right)} d x}}{4} + \frac{\int{\sin{\left(2 x \right)} \sin{\left(5 x \right)} \cos{\left(7 x \right)} d x}}{2} + \frac{{\color{red}{\left(\int{1 d u} - \int{\cos{\left(2 u \right)} d u}\right)}}}{40}$$

配合 $$$c=1$$$，應用常數法則 $$$\int c\, du = c u$$$：

$$- \frac{\int{\sin{\left(x \right)} \sin{\left(5 x \right)} d x}}{4} + \frac{\int{\sin{\left(2 x \right)} \sin{\left(5 x \right)} \cos{\left(7 x \right)} d x}}{2} - \frac{\int{\cos{\left(2 u \right)} d u}}{40} + \frac{{\color{red}{\int{1 d u}}}}{40} = - \frac{\int{\sin{\left(x \right)} \sin{\left(5 x \right)} d x}}{4} + \frac{\int{\sin{\left(2 x \right)} \sin{\left(5 x \right)} \cos{\left(7 x \right)} d x}}{2} - \frac{\int{\cos{\left(2 u \right)} d u}}{40} + \frac{{\color{red}{u}}}{40}$$

令 $$$v=2 u$$$。

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

該積分變為

$$\frac{u}{40} - \frac{\int{\sin{\left(x \right)} \sin{\left(5 x \right)} d x}}{4} + \frac{\int{\sin{\left(2 x \right)} \sin{\left(5 x \right)} \cos{\left(7 x \right)} d x}}{2} - \frac{{\color{red}{\int{\cos{\left(2 u \right)} d u}}}}{40} = \frac{u}{40} - \frac{\int{\sin{\left(x \right)} \sin{\left(5 x \right)} d x}}{4} + \frac{\int{\sin{\left(2 x \right)} \sin{\left(5 x \right)} \cos{\left(7 x \right)} d x}}{2} - \frac{{\color{red}{\int{\frac{\cos{\left(v \right)}}{2} d v}}}}{40}$$

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

$$\frac{u}{40} - \frac{\int{\sin{\left(x \right)} \sin{\left(5 x \right)} d x}}{4} + \frac{\int{\sin{\left(2 x \right)} \sin{\left(5 x \right)} \cos{\left(7 x \right)} d x}}{2} - \frac{{\color{red}{\int{\frac{\cos{\left(v \right)}}{2} d v}}}}{40} = \frac{u}{40} - \frac{\int{\sin{\left(x \right)} \sin{\left(5 x \right)} d x}}{4} + \frac{\int{\sin{\left(2 x \right)} \sin{\left(5 x \right)} \cos{\left(7 x \right)} d x}}{2} - \frac{{\color{red}{\left(\frac{\int{\cos{\left(v \right)} d v}}{2}\right)}}}{40}$$

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

$$\frac{u}{40} - \frac{\int{\sin{\left(x \right)} \sin{\left(5 x \right)} d x}}{4} + \frac{\int{\sin{\left(2 x \right)} \sin{\left(5 x \right)} \cos{\left(7 x \right)} d x}}{2} - \frac{{\color{red}{\int{\cos{\left(v \right)} d v}}}}{80} = \frac{u}{40} - \frac{\int{\sin{\left(x \right)} \sin{\left(5 x \right)} d x}}{4} + \frac{\int{\sin{\left(2 x \right)} \sin{\left(5 x \right)} \cos{\left(7 x \right)} d x}}{2} - \frac{{\color{red}{\sin{\left(v \right)}}}}{80}$$

回顧一下 $$$v=2 u$$$：

$$\frac{u}{40} - \frac{\int{\sin{\left(x \right)} \sin{\left(5 x \right)} d x}}{4} + \frac{\int{\sin{\left(2 x \right)} \sin{\left(5 x \right)} \cos{\left(7 x \right)} d x}}{2} - \frac{\sin{\left({\color{red}{v}} \right)}}{80} = \frac{u}{40} - \frac{\int{\sin{\left(x \right)} \sin{\left(5 x \right)} d x}}{4} + \frac{\int{\sin{\left(2 x \right)} \sin{\left(5 x \right)} \cos{\left(7 x \right)} d x}}{2} - \frac{\sin{\left({\color{red}{\left(2 u\right)}} \right)}}{80}$$

回顧一下 $$$u=5 x$$$：

$$- \frac{\int{\sin{\left(x \right)} \sin{\left(5 x \right)} d x}}{4} + \frac{\int{\sin{\left(2 x \right)} \sin{\left(5 x \right)} \cos{\left(7 x \right)} d x}}{2} - \frac{\sin{\left(2 {\color{red}{u}} \right)}}{80} + \frac{{\color{red}{u}}}{40} = - \frac{\int{\sin{\left(x \right)} \sin{\left(5 x \right)} d x}}{4} + \frac{\int{\sin{\left(2 x \right)} \sin{\left(5 x \right)} \cos{\left(7 x \right)} d x}}{2} - \frac{\sin{\left(2 {\color{red}{\left(5 x\right)}} \right)}}{80} + \frac{{\color{red}{\left(5 x\right)}}}{40}$$

使用公式 $$$\sin\left(\alpha \right)\sin\left(\beta \right)=\frac{1}{2} \cos\left(\alpha-\beta \right)-\frac{1}{2} \cos\left(\alpha+\beta \right)$$$，並配合 $$$\alpha=x$$$ 與 $$$\beta=5 x$$$，改寫被積分函數:

$$\frac{x}{8} - \frac{\sin{\left(10 x \right)}}{80} + \frac{\int{\sin{\left(2 x \right)} \sin{\left(5 x \right)} \cos{\left(7 x \right)} d x}}{2} - \frac{{\color{red}{\int{\sin{\left(x \right)} \sin{\left(5 x \right)} d x}}}}{4} = \frac{x}{8} - \frac{\sin{\left(10 x \right)}}{80} + \frac{\int{\sin{\left(2 x \right)} \sin{\left(5 x \right)} \cos{\left(7 x \right)} d x}}{2} - \frac{{\color{red}{\int{\left(\frac{\cos{\left(4 x \right)}}{2} - \frac{\cos{\left(6 x \right)}}{2}\right)d x}}}}{4}$$

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 $$$f{\left(x \right)} = \cos{\left(4 x \right)} - \cos{\left(6 x \right)}$$$：

$$\frac{x}{8} - \frac{\sin{\left(10 x \right)}}{80} + \frac{\int{\sin{\left(2 x \right)} \sin{\left(5 x \right)} \cos{\left(7 x \right)} d x}}{2} - \frac{{\color{red}{\int{\left(\frac{\cos{\left(4 x \right)}}{2} - \frac{\cos{\left(6 x \right)}}{2}\right)d x}}}}{4} = \frac{x}{8} - \frac{\sin{\left(10 x \right)}}{80} + \frac{\int{\sin{\left(2 x \right)} \sin{\left(5 x \right)} \cos{\left(7 x \right)} d x}}{2} - \frac{{\color{red}{\left(\frac{\int{\left(\cos{\left(4 x \right)} - \cos{\left(6 x \right)}\right)d x}}{2}\right)}}}{4}$$

逐項積分：

$$\frac{x}{8} - \frac{\sin{\left(10 x \right)}}{80} + \frac{\int{\sin{\left(2 x \right)} \sin{\left(5 x \right)} \cos{\left(7 x \right)} d x}}{2} - \frac{{\color{red}{\int{\left(\cos{\left(4 x \right)} - \cos{\left(6 x \right)}\right)d x}}}}{8} = \frac{x}{8} - \frac{\sin{\left(10 x \right)}}{80} + \frac{\int{\sin{\left(2 x \right)} \sin{\left(5 x \right)} \cos{\left(7 x \right)} d x}}{2} - \frac{{\color{red}{\left(\int{\cos{\left(4 x \right)} d x} - \int{\cos{\left(6 x \right)} d x}\right)}}}{8}$$

令 $$$u=6 x$$$。

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

因此，

$$\frac{x}{8} - \frac{\sin{\left(10 x \right)}}{80} + \frac{\int{\sin{\left(2 x \right)} \sin{\left(5 x \right)} \cos{\left(7 x \right)} d x}}{2} - \frac{\int{\cos{\left(4 x \right)} d x}}{8} + \frac{{\color{red}{\int{\cos{\left(6 x \right)} d x}}}}{8} = \frac{x}{8} - \frac{\sin{\left(10 x \right)}}{80} + \frac{\int{\sin{\left(2 x \right)} \sin{\left(5 x \right)} \cos{\left(7 x \right)} d x}}{2} - \frac{\int{\cos{\left(4 x \right)} d x}}{8} + \frac{{\color{red}{\int{\frac{\cos{\left(u \right)}}{6} d u}}}}{8}$$

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

$$\frac{x}{8} - \frac{\sin{\left(10 x \right)}}{80} + \frac{\int{\sin{\left(2 x \right)} \sin{\left(5 x \right)} \cos{\left(7 x \right)} d x}}{2} - \frac{\int{\cos{\left(4 x \right)} d x}}{8} + \frac{{\color{red}{\int{\frac{\cos{\left(u \right)}}{6} d u}}}}{8} = \frac{x}{8} - \frac{\sin{\left(10 x \right)}}{80} + \frac{\int{\sin{\left(2 x \right)} \sin{\left(5 x \right)} \cos{\left(7 x \right)} d x}}{2} - \frac{\int{\cos{\left(4 x \right)} d x}}{8} + \frac{{\color{red}{\left(\frac{\int{\cos{\left(u \right)} d u}}{6}\right)}}}{8}$$

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

$$\frac{x}{8} - \frac{\sin{\left(10 x \right)}}{80} + \frac{\int{\sin{\left(2 x \right)} \sin{\left(5 x \right)} \cos{\left(7 x \right)} d x}}{2} - \frac{\int{\cos{\left(4 x \right)} d x}}{8} + \frac{{\color{red}{\int{\cos{\left(u \right)} d u}}}}{48} = \frac{x}{8} - \frac{\sin{\left(10 x \right)}}{80} + \frac{\int{\sin{\left(2 x \right)} \sin{\left(5 x \right)} \cos{\left(7 x \right)} d x}}{2} - \frac{\int{\cos{\left(4 x \right)} d x}}{8} + \frac{{\color{red}{\sin{\left(u \right)}}}}{48}$$

回顧一下 $$$u=6 x$$$：

$$\frac{x}{8} - \frac{\sin{\left(10 x \right)}}{80} + \frac{\int{\sin{\left(2 x \right)} \sin{\left(5 x \right)} \cos{\left(7 x \right)} d x}}{2} - \frac{\int{\cos{\left(4 x \right)} d x}}{8} + \frac{\sin{\left({\color{red}{u}} \right)}}{48} = \frac{x}{8} - \frac{\sin{\left(10 x \right)}}{80} + \frac{\int{\sin{\left(2 x \right)} \sin{\left(5 x \right)} \cos{\left(7 x \right)} d x}}{2} - \frac{\int{\cos{\left(4 x \right)} d x}}{8} + \frac{\sin{\left({\color{red}{\left(6 x\right)}} \right)}}{48}$$

令 $$$u=4 x$$$。

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

該積分可改寫為

$$\frac{x}{8} + \frac{\sin{\left(6 x \right)}}{48} - \frac{\sin{\left(10 x \right)}}{80} + \frac{\int{\sin{\left(2 x \right)} \sin{\left(5 x \right)} \cos{\left(7 x \right)} d x}}{2} - \frac{{\color{red}{\int{\cos{\left(4 x \right)} d x}}}}{8} = \frac{x}{8} + \frac{\sin{\left(6 x \right)}}{48} - \frac{\sin{\left(10 x \right)}}{80} + \frac{\int{\sin{\left(2 x \right)} \sin{\left(5 x \right)} \cos{\left(7 x \right)} d x}}{2} - \frac{{\color{red}{\int{\frac{\cos{\left(u \right)}}{4} d u}}}}{8}$$

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

$$\frac{x}{8} + \frac{\sin{\left(6 x \right)}}{48} - \frac{\sin{\left(10 x \right)}}{80} + \frac{\int{\sin{\left(2 x \right)} \sin{\left(5 x \right)} \cos{\left(7 x \right)} d x}}{2} - \frac{{\color{red}{\int{\frac{\cos{\left(u \right)}}{4} d u}}}}{8} = \frac{x}{8} + \frac{\sin{\left(6 x \right)}}{48} - \frac{\sin{\left(10 x \right)}}{80} + \frac{\int{\sin{\left(2 x \right)} \sin{\left(5 x \right)} \cos{\left(7 x \right)} d x}}{2} - \frac{{\color{red}{\left(\frac{\int{\cos{\left(u \right)} d u}}{4}\right)}}}{8}$$

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

$$\frac{x}{8} + \frac{\sin{\left(6 x \right)}}{48} - \frac{\sin{\left(10 x \right)}}{80} + \frac{\int{\sin{\left(2 x \right)} \sin{\left(5 x \right)} \cos{\left(7 x \right)} d x}}{2} - \frac{{\color{red}{\int{\cos{\left(u \right)} d u}}}}{32} = \frac{x}{8} + \frac{\sin{\left(6 x \right)}}{48} - \frac{\sin{\left(10 x \right)}}{80} + \frac{\int{\sin{\left(2 x \right)} \sin{\left(5 x \right)} \cos{\left(7 x \right)} d x}}{2} - \frac{{\color{red}{\sin{\left(u \right)}}}}{32}$$

回顧一下 $$$u=4 x$$$：

$$\frac{x}{8} + \frac{\sin{\left(6 x \right)}}{48} - \frac{\sin{\left(10 x \right)}}{80} + \frac{\int{\sin{\left(2 x \right)} \sin{\left(5 x \right)} \cos{\left(7 x \right)} d x}}{2} - \frac{\sin{\left({\color{red}{u}} \right)}}{32} = \frac{x}{8} + \frac{\sin{\left(6 x \right)}}{48} - \frac{\sin{\left(10 x \right)}}{80} + \frac{\int{\sin{\left(2 x \right)} \sin{\left(5 x \right)} \cos{\left(7 x \right)} d x}}{2} - \frac{\sin{\left({\color{red}{\left(4 x\right)}} \right)}}{32}$$

使用公式 $$$\sin\left(\alpha \right)\cos\left(\beta \right)=\frac{1}{2} \sin\left(\alpha-\beta \right)+\frac{1}{2} \sin\left(\alpha+\beta \right)$$$，以 $$$\alpha=2 x$$$ 和 $$$\beta=7 x$$$ 將 $$$\sin\left(2 x \right)\cos\left(7 x \right)$$$ 改寫:

$$\frac{x}{8} - \frac{\sin{\left(4 x \right)}}{32} + \frac{\sin{\left(6 x \right)}}{48} - \frac{\sin{\left(10 x \right)}}{80} + \frac{{\color{red}{\int{\sin{\left(2 x \right)} \sin{\left(5 x \right)} \cos{\left(7 x \right)} d x}}}}{2} = \frac{x}{8} - \frac{\sin{\left(4 x \right)}}{32} + \frac{\sin{\left(6 x \right)}}{48} - \frac{\sin{\left(10 x \right)}}{80} + \frac{{\color{red}{\int{\left(- \frac{\sin{\left(5 x \right)}}{2} + \frac{\sin{\left(9 x \right)}}{2}\right) \sin{\left(5 x \right)} d x}}}}{2}$$

展開該表達式:

$$\frac{x}{8} - \frac{\sin{\left(4 x \right)}}{32} + \frac{\sin{\left(6 x \right)}}{48} - \frac{\sin{\left(10 x \right)}}{80} + \frac{{\color{red}{\int{\left(- \frac{\sin{\left(5 x \right)}}{2} + \frac{\sin{\left(9 x \right)}}{2}\right) \sin{\left(5 x \right)} d x}}}}{2} = \frac{x}{8} - \frac{\sin{\left(4 x \right)}}{32} + \frac{\sin{\left(6 x \right)}}{48} - \frac{\sin{\left(10 x \right)}}{80} + \frac{{\color{red}{\int{\left(- \frac{\sin^{2}{\left(5 x \right)}}{2} + \frac{\sin{\left(5 x \right)} \sin{\left(9 x \right)}}{2}\right)d x}}}}{2}$$

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 $$$f{\left(x \right)} = - \sin^{2}{\left(5 x \right)} + \sin{\left(5 x \right)} \sin{\left(9 x \right)}$$$：

$$\frac{x}{8} - \frac{\sin{\left(4 x \right)}}{32} + \frac{\sin{\left(6 x \right)}}{48} - \frac{\sin{\left(10 x \right)}}{80} + \frac{{\color{red}{\int{\left(- \frac{\sin^{2}{\left(5 x \right)}}{2} + \frac{\sin{\left(5 x \right)} \sin{\left(9 x \right)}}{2}\right)d x}}}}{2} = \frac{x}{8} - \frac{\sin{\left(4 x \right)}}{32} + \frac{\sin{\left(6 x \right)}}{48} - \frac{\sin{\left(10 x \right)}}{80} + \frac{{\color{red}{\left(\frac{\int{\left(- \sin^{2}{\left(5 x \right)} + \sin{\left(5 x \right)} \sin{\left(9 x \right)}\right)d x}}{2}\right)}}}{2}$$

逐項積分：

$$\frac{x}{8} - \frac{\sin{\left(4 x \right)}}{32} + \frac{\sin{\left(6 x \right)}}{48} - \frac{\sin{\left(10 x \right)}}{80} + \frac{{\color{red}{\int{\left(- \sin^{2}{\left(5 x \right)} + \sin{\left(5 x \right)} \sin{\left(9 x \right)}\right)d x}}}}{4} = \frac{x}{8} - \frac{\sin{\left(4 x \right)}}{32} + \frac{\sin{\left(6 x \right)}}{48} - \frac{\sin{\left(10 x \right)}}{80} + \frac{{\color{red}{\left(\int{\sin{\left(5 x \right)} \sin{\left(9 x \right)} d x} - \int{\sin^{2}{\left(5 x \right)} d x}\right)}}}{4}$$

積分 $$$\int{\sin^{2}{\left(5 x \right)} d x}$$$ 已經計算過：

$$\int{\sin^{2}{\left(5 x \right)} d x} = \frac{x}{2} - \frac{\sin{\left(10 x \right)}}{20}$$

因此，

$$\frac{x}{8} - \frac{\sin{\left(4 x \right)}}{32} + \frac{\sin{\left(6 x \right)}}{48} - \frac{\sin{\left(10 x \right)}}{80} + \frac{\int{\sin{\left(5 x \right)} \sin{\left(9 x \right)} d x}}{4} - \frac{{\color{red}{\int{\sin^{2}{\left(5 x \right)} d x}}}}{4} = \frac{x}{8} - \frac{\sin{\left(4 x \right)}}{32} + \frac{\sin{\left(6 x \right)}}{48} - \frac{\sin{\left(10 x \right)}}{80} + \frac{\int{\sin{\left(5 x \right)} \sin{\left(9 x \right)} d x}}{4} - \frac{{\color{red}{\left(\frac{x}{2} - \frac{\sin{\left(10 x \right)}}{20}\right)}}}{4}$$

使用公式 $$$\sin\left(\alpha \right)\sin\left(\beta \right)=\frac{1}{2} \cos\left(\alpha-\beta \right)-\frac{1}{2} \cos\left(\alpha+\beta \right)$$$，並配合 $$$\alpha=5 x$$$ 與 $$$\beta=9 x$$$，改寫被積分函數:

$$- \frac{\sin{\left(4 x \right)}}{32} + \frac{\sin{\left(6 x \right)}}{48} + \frac{{\color{red}{\int{\sin{\left(5 x \right)} \sin{\left(9 x \right)} d x}}}}{4} = - \frac{\sin{\left(4 x \right)}}{32} + \frac{\sin{\left(6 x \right)}}{48} + \frac{{\color{red}{\int{\left(\frac{\cos{\left(4 x \right)}}{2} - \frac{\cos{\left(14 x \right)}}{2}\right)d x}}}}{4}$$

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 $$$f{\left(x \right)} = \cos{\left(4 x \right)} - \cos{\left(14 x \right)}$$$：

$$- \frac{\sin{\left(4 x \right)}}{32} + \frac{\sin{\left(6 x \right)}}{48} + \frac{{\color{red}{\int{\left(\frac{\cos{\left(4 x \right)}}{2} - \frac{\cos{\left(14 x \right)}}{2}\right)d x}}}}{4} = - \frac{\sin{\left(4 x \right)}}{32} + \frac{\sin{\left(6 x \right)}}{48} + \frac{{\color{red}{\left(\frac{\int{\left(\cos{\left(4 x \right)} - \cos{\left(14 x \right)}\right)d x}}{2}\right)}}}{4}$$

逐項積分：

$$- \frac{\sin{\left(4 x \right)}}{32} + \frac{\sin{\left(6 x \right)}}{48} + \frac{{\color{red}{\int{\left(\cos{\left(4 x \right)} - \cos{\left(14 x \right)}\right)d x}}}}{8} = - \frac{\sin{\left(4 x \right)}}{32} + \frac{\sin{\left(6 x \right)}}{48} + \frac{{\color{red}{\left(\int{\cos{\left(4 x \right)} d x} - \int{\cos{\left(14 x \right)} d x}\right)}}}{8}$$

令 $$$v=14 x$$$。

則 $$$dv=\left(14 x\right)^{\prime }dx = 14 dx$$$ (步驟見»)，並可得 $$$dx = \frac{dv}{14}$$$。

因此，

$$- \frac{\sin{\left(4 x \right)}}{32} + \frac{\sin{\left(6 x \right)}}{48} + \frac{\int{\cos{\left(4 x \right)} d x}}{8} - \frac{{\color{red}{\int{\cos{\left(14 x \right)} d x}}}}{8} = - \frac{\sin{\left(4 x \right)}}{32} + \frac{\sin{\left(6 x \right)}}{48} + \frac{\int{\cos{\left(4 x \right)} d x}}{8} - \frac{{\color{red}{\int{\frac{\cos{\left(v \right)}}{14} d v}}}}{8}$$

套用常數倍法則 $$$\int c f{\left(v \right)}\, dv = c \int f{\left(v \right)}\, dv$$$，使用 $$$c=\frac{1}{14}$$$ 與 $$$f{\left(v \right)} = \cos{\left(v \right)}$$$：

$$- \frac{\sin{\left(4 x \right)}}{32} + \frac{\sin{\left(6 x \right)}}{48} + \frac{\int{\cos{\left(4 x \right)} d x}}{8} - \frac{{\color{red}{\int{\frac{\cos{\left(v \right)}}{14} d v}}}}{8} = - \frac{\sin{\left(4 x \right)}}{32} + \frac{\sin{\left(6 x \right)}}{48} + \frac{\int{\cos{\left(4 x \right)} d x}}{8} - \frac{{\color{red}{\left(\frac{\int{\cos{\left(v \right)} d v}}{14}\right)}}}{8}$$

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

$$- \frac{\sin{\left(4 x \right)}}{32} + \frac{\sin{\left(6 x \right)}}{48} + \frac{\int{\cos{\left(4 x \right)} d x}}{8} - \frac{{\color{red}{\int{\cos{\left(v \right)} d v}}}}{112} = - \frac{\sin{\left(4 x \right)}}{32} + \frac{\sin{\left(6 x \right)}}{48} + \frac{\int{\cos{\left(4 x \right)} d x}}{8} - \frac{{\color{red}{\sin{\left(v \right)}}}}{112}$$

回顧一下 $$$v=14 x$$$：

$$- \frac{\sin{\left(4 x \right)}}{32} + \frac{\sin{\left(6 x \right)}}{48} + \frac{\int{\cos{\left(4 x \right)} d x}}{8} - \frac{\sin{\left({\color{red}{v}} \right)}}{112} = - \frac{\sin{\left(4 x \right)}}{32} + \frac{\sin{\left(6 x \right)}}{48} + \frac{\int{\cos{\left(4 x \right)} d x}}{8} - \frac{\sin{\left({\color{red}{\left(14 x\right)}} \right)}}{112}$$

積分 $$$\int{\cos{\left(4 x \right)} d x}$$$ 已經計算過：

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

因此，

$$- \frac{\sin{\left(4 x \right)}}{32} + \frac{\sin{\left(6 x \right)}}{48} - \frac{\sin{\left(14 x \right)}}{112} + \frac{{\color{red}{\int{\cos{\left(4 x \right)} d x}}}}{8} = - \frac{\sin{\left(4 x \right)}}{32} + \frac{\sin{\left(6 x \right)}}{48} - \frac{\sin{\left(14 x \right)}}{112} + \frac{{\color{red}{\left(\frac{\sin{\left(4 x \right)}}{4}\right)}}}{8}$$

因此，

$$\int{\sin{\left(2 x \right)} \sin{\left(5 x \right)} \cos{\left(2 x \right)} \cos{\left(5 x \right)} d x} = \frac{\sin{\left(6 x \right)}}{48} - \frac{\sin{\left(14 x \right)}}{112}$$

加上積分常數：

$$\int{\sin{\left(2 x \right)} \sin{\left(5 x \right)} \cos{\left(2 x \right)} \cos{\left(5 x \right)} d x} = \frac{\sin{\left(6 x \right)}}{48} - \frac{\sin{\left(14 x \right)}}{112}+C$$

$$$\int \sin{\left(2 x \right)} \sin{\left(5 x \right)} \cos{\left(2 x \right)} \cos{\left(5 x \right)}\, dx = \left(\frac{\sin{\left(6 x \right)}}{48} - \frac{\sin{\left(14 x \right)}}{112}\right) + C$$$A