$$$5 \sin^{2}{\left(7 x \right)} \cos^{3}{\left(7 x \right)}$$$ 的積分

求$$$\int 5 \sin^{2}{\left(7 x \right)} \cos^{3}{\left(7 x \right)}\, dx$$$。

套用降冪公式 $$$\cos^{3}{\left(\alpha \right)} = \frac{3 \cos{\left(\alpha \right)}}{4} + \frac{\cos{\left(3 \alpha \right)}}{4}$$$，令 $$$\alpha=7 x$$$:

$${\color{red}{\int{5 \sin^{2}{\left(7 x \right)} \cos^{3}{\left(7 x \right)} d x}}} = {\color{red}{\int{\frac{5 \left(3 \cos{\left(7 x \right)} + \cos{\left(21 x \right)}\right) \sin^{2}{\left(7 x \right)}}{4} d x}}}$$

套用降冪公式 $$$\sin^{2}{\left(\alpha \right)} = \frac{1}{2} - \frac{\cos{\left(2 \alpha \right)}}{2}$$$，令 $$$\alpha=7 x$$$:

$${\color{red}{\int{\frac{5 \left(3 \cos{\left(7 x \right)} + \cos{\left(21 x \right)}\right) \sin^{2}{\left(7 x \right)}}{4} d x}}} = {\color{red}{\int{\frac{5 \left(1 - \cos{\left(14 x \right)}\right) \left(3 \cos{\left(7 x \right)} + \cos{\left(21 x \right)}\right)}{8} d x}}}$$

套用常數倍法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$，使用 $$$c=\frac{1}{8}$$$ 與 $$$f{\left(x \right)} = 5 \left(1 - \cos{\left(14 x \right)}\right) \left(3 \cos{\left(7 x \right)} + \cos{\left(21 x \right)}\right)$$$：

$${\color{red}{\int{\frac{5 \left(1 - \cos{\left(14 x \right)}\right) \left(3 \cos{\left(7 x \right)} + \cos{\left(21 x \right)}\right)}{8} d x}}} = {\color{red}{\left(\frac{\int{5 \left(1 - \cos{\left(14 x \right)}\right) \left(3 \cos{\left(7 x \right)} + \cos{\left(21 x \right)}\right) d x}}{8}\right)}}$$

Expand the expression:

$$\frac{{\color{red}{\int{5 \left(1 - \cos{\left(14 x \right)}\right) \left(3 \cos{\left(7 x \right)} + \cos{\left(21 x \right)}\right) d x}}}}{8} = \frac{{\color{red}{\int{\left(- 15 \cos{\left(7 x \right)} \cos{\left(14 x \right)} + 15 \cos{\left(7 x \right)} - 5 \cos{\left(14 x \right)} \cos{\left(21 x \right)} + 5 \cos{\left(21 x \right)}\right)d x}}}}{8}$$

逐項積分：

$$\frac{{\color{red}{\int{\left(- 15 \cos{\left(7 x \right)} \cos{\left(14 x \right)} + 15 \cos{\left(7 x \right)} - 5 \cos{\left(14 x \right)} \cos{\left(21 x \right)} + 5 \cos{\left(21 x \right)}\right)d x}}}}{8} = \frac{{\color{red}{\left(- \int{15 \cos{\left(7 x \right)} \cos{\left(14 x \right)} d x} - \int{5 \cos{\left(14 x \right)} \cos{\left(21 x \right)} d x} + \int{15 \cos{\left(7 x \right)} d x} + \int{5 \cos{\left(21 x \right)} d x}\right)}}}{8}$$

套用常數倍法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$，使用 $$$c=5$$$ 與 $$$f{\left(x \right)} = \cos{\left(21 x \right)}$$$：

$$- \frac{\int{15 \cos{\left(7 x \right)} \cos{\left(14 x \right)} d x}}{8} - \frac{\int{5 \cos{\left(14 x \right)} \cos{\left(21 x \right)} d x}}{8} + \frac{\int{15 \cos{\left(7 x \right)} d x}}{8} + \frac{{\color{red}{\int{5 \cos{\left(21 x \right)} d x}}}}{8} = - \frac{\int{15 \cos{\left(7 x \right)} \cos{\left(14 x \right)} d x}}{8} - \frac{\int{5 \cos{\left(14 x \right)} \cos{\left(21 x \right)} d x}}{8} + \frac{\int{15 \cos{\left(7 x \right)} d x}}{8} + \frac{{\color{red}{\left(5 \int{\cos{\left(21 x \right)} d x}\right)}}}{8}$$

令 $$$u=21 x$$$。

則 $$$du=\left(21 x\right)^{\prime }dx = 21 dx$$$ (步驟見»)，並可得 $$$dx = \frac{du}{21}$$$。

該積分變為

$$- \frac{\int{15 \cos{\left(7 x \right)} \cos{\left(14 x \right)} d x}}{8} - \frac{\int{5 \cos{\left(14 x \right)} \cos{\left(21 x \right)} d x}}{8} + \frac{\int{15 \cos{\left(7 x \right)} d x}}{8} + \frac{5 {\color{red}{\int{\cos{\left(21 x \right)} d x}}}}{8} = - \frac{\int{15 \cos{\left(7 x \right)} \cos{\left(14 x \right)} d x}}{8} - \frac{\int{5 \cos{\left(14 x \right)} \cos{\left(21 x \right)} d x}}{8} + \frac{\int{15 \cos{\left(7 x \right)} d x}}{8} + \frac{5 {\color{red}{\int{\frac{\cos{\left(u \right)}}{21} d u}}}}{8}$$

套用常數倍法則 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$，使用 $$$c=\frac{1}{21}$$$ 與 $$$f{\left(u \right)} = \cos{\left(u \right)}$$$：

$$- \frac{\int{15 \cos{\left(7 x \right)} \cos{\left(14 x \right)} d x}}{8} - \frac{\int{5 \cos{\left(14 x \right)} \cos{\left(21 x \right)} d x}}{8} + \frac{\int{15 \cos{\left(7 x \right)} d x}}{8} + \frac{5 {\color{red}{\int{\frac{\cos{\left(u \right)}}{21} d u}}}}{8} = - \frac{\int{15 \cos{\left(7 x \right)} \cos{\left(14 x \right)} d x}}{8} - \frac{\int{5 \cos{\left(14 x \right)} \cos{\left(21 x \right)} d x}}{8} + \frac{\int{15 \cos{\left(7 x \right)} d x}}{8} + \frac{5 {\color{red}{\left(\frac{\int{\cos{\left(u \right)} d u}}{21}\right)}}}{8}$$

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

$$- \frac{\int{15 \cos{\left(7 x \right)} \cos{\left(14 x \right)} d x}}{8} - \frac{\int{5 \cos{\left(14 x \right)} \cos{\left(21 x \right)} d x}}{8} + \frac{\int{15 \cos{\left(7 x \right)} d x}}{8} + \frac{5 {\color{red}{\int{\cos{\left(u \right)} d u}}}}{168} = - \frac{\int{15 \cos{\left(7 x \right)} \cos{\left(14 x \right)} d x}}{8} - \frac{\int{5 \cos{\left(14 x \right)} \cos{\left(21 x \right)} d x}}{8} + \frac{\int{15 \cos{\left(7 x \right)} d x}}{8} + \frac{5 {\color{red}{\sin{\left(u \right)}}}}{168}$$

回顧一下 $$$u=21 x$$$：

$$- \frac{\int{15 \cos{\left(7 x \right)} \cos{\left(14 x \right)} d x}}{8} - \frac{\int{5 \cos{\left(14 x \right)} \cos{\left(21 x \right)} d x}}{8} + \frac{\int{15 \cos{\left(7 x \right)} d x}}{8} + \frac{5 \sin{\left({\color{red}{u}} \right)}}{168} = - \frac{\int{15 \cos{\left(7 x \right)} \cos{\left(14 x \right)} d x}}{8} - \frac{\int{5 \cos{\left(14 x \right)} \cos{\left(21 x \right)} d x}}{8} + \frac{\int{15 \cos{\left(7 x \right)} d x}}{8} + \frac{5 \sin{\left({\color{red}{\left(21 x\right)}} \right)}}{168}$$

套用常數倍法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$，使用 $$$c=15$$$ 與 $$$f{\left(x \right)} = \cos{\left(7 x \right)}$$$：

$$\frac{5 \sin{\left(21 x \right)}}{168} - \frac{\int{15 \cos{\left(7 x \right)} \cos{\left(14 x \right)} d x}}{8} - \frac{\int{5 \cos{\left(14 x \right)} \cos{\left(21 x \right)} d x}}{8} + \frac{{\color{red}{\int{15 \cos{\left(7 x \right)} d x}}}}{8} = \frac{5 \sin{\left(21 x \right)}}{168} - \frac{\int{15 \cos{\left(7 x \right)} \cos{\left(14 x \right)} d x}}{8} - \frac{\int{5 \cos{\left(14 x \right)} \cos{\left(21 x \right)} d x}}{8} + \frac{{\color{red}{\left(15 \int{\cos{\left(7 x \right)} d x}\right)}}}{8}$$

令 $$$u=7 x$$$。

則 $$$du=\left(7 x\right)^{\prime }dx = 7 dx$$$ (步驟見»)，並可得 $$$dx = \frac{du}{7}$$$。

該積分可改寫為

$$\frac{5 \sin{\left(21 x \right)}}{168} - \frac{\int{15 \cos{\left(7 x \right)} \cos{\left(14 x \right)} d x}}{8} - \frac{\int{5 \cos{\left(14 x \right)} \cos{\left(21 x \right)} d x}}{8} + \frac{15 {\color{red}{\int{\cos{\left(7 x \right)} d x}}}}{8} = \frac{5 \sin{\left(21 x \right)}}{168} - \frac{\int{15 \cos{\left(7 x \right)} \cos{\left(14 x \right)} d x}}{8} - \frac{\int{5 \cos{\left(14 x \right)} \cos{\left(21 x \right)} d x}}{8} + \frac{15 {\color{red}{\int{\frac{\cos{\left(u \right)}}{7} d u}}}}{8}$$

套用常數倍法則 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$，使用 $$$c=\frac{1}{7}$$$ 與 $$$f{\left(u \right)} = \cos{\left(u \right)}$$$：

$$\frac{5 \sin{\left(21 x \right)}}{168} - \frac{\int{15 \cos{\left(7 x \right)} \cos{\left(14 x \right)} d x}}{8} - \frac{\int{5 \cos{\left(14 x \right)} \cos{\left(21 x \right)} d x}}{8} + \frac{15 {\color{red}{\int{\frac{\cos{\left(u \right)}}{7} d u}}}}{8} = \frac{5 \sin{\left(21 x \right)}}{168} - \frac{\int{15 \cos{\left(7 x \right)} \cos{\left(14 x \right)} d x}}{8} - \frac{\int{5 \cos{\left(14 x \right)} \cos{\left(21 x \right)} d x}}{8} + \frac{15 {\color{red}{\left(\frac{\int{\cos{\left(u \right)} d u}}{7}\right)}}}{8}$$

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

$$\frac{5 \sin{\left(21 x \right)}}{168} - \frac{\int{15 \cos{\left(7 x \right)} \cos{\left(14 x \right)} d x}}{8} - \frac{\int{5 \cos{\left(14 x \right)} \cos{\left(21 x \right)} d x}}{8} + \frac{15 {\color{red}{\int{\cos{\left(u \right)} d u}}}}{56} = \frac{5 \sin{\left(21 x \right)}}{168} - \frac{\int{15 \cos{\left(7 x \right)} \cos{\left(14 x \right)} d x}}{8} - \frac{\int{5 \cos{\left(14 x \right)} \cos{\left(21 x \right)} d x}}{8} + \frac{15 {\color{red}{\sin{\left(u \right)}}}}{56}$$

回顧一下 $$$u=7 x$$$：

$$\frac{5 \sin{\left(21 x \right)}}{168} - \frac{\int{15 \cos{\left(7 x \right)} \cos{\left(14 x \right)} d x}}{8} - \frac{\int{5 \cos{\left(14 x \right)} \cos{\left(21 x \right)} d x}}{8} + \frac{15 \sin{\left({\color{red}{u}} \right)}}{56} = \frac{5 \sin{\left(21 x \right)}}{168} - \frac{\int{15 \cos{\left(7 x \right)} \cos{\left(14 x \right)} d x}}{8} - \frac{\int{5 \cos{\left(14 x \right)} \cos{\left(21 x \right)} d x}}{8} + \frac{15 \sin{\left({\color{red}{\left(7 x\right)}} \right)}}{56}$$

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 $$$\alpha=7 x$$$ 和 $$$\beta=14 x$$$ 將 $$$\cos\left(7 x \right)\cos\left(14 x \right)$$$ 改寫:

$$\frac{15 \sin{\left(7 x \right)}}{56} + \frac{5 \sin{\left(21 x \right)}}{168} - \frac{\int{5 \cos{\left(14 x \right)} \cos{\left(21 x \right)} d x}}{8} - \frac{{\color{red}{\int{15 \cos{\left(7 x \right)} \cos{\left(14 x \right)} d x}}}}{8} = \frac{15 \sin{\left(7 x \right)}}{56} + \frac{5 \sin{\left(21 x \right)}}{168} - \frac{\int{5 \cos{\left(14 x \right)} \cos{\left(21 x \right)} d x}}{8} - \frac{{\color{red}{\int{\left(\frac{15 \cos{\left(7 x \right)}}{2} + \frac{15 \cos{\left(21 x \right)}}{2}\right)d x}}}}{8}$$

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 $$$f{\left(x \right)} = 15 \cos{\left(7 x \right)} + 15 \cos{\left(21 x \right)}$$$：

$$\frac{15 \sin{\left(7 x \right)}}{56} + \frac{5 \sin{\left(21 x \right)}}{168} - \frac{\int{5 \cos{\left(14 x \right)} \cos{\left(21 x \right)} d x}}{8} - \frac{{\color{red}{\int{\left(\frac{15 \cos{\left(7 x \right)}}{2} + \frac{15 \cos{\left(21 x \right)}}{2}\right)d x}}}}{8} = \frac{15 \sin{\left(7 x \right)}}{56} + \frac{5 \sin{\left(21 x \right)}}{168} - \frac{\int{5 \cos{\left(14 x \right)} \cos{\left(21 x \right)} d x}}{8} - \frac{{\color{red}{\left(\frac{\int{\left(15 \cos{\left(7 x \right)} + 15 \cos{\left(21 x \right)}\right)d x}}{2}\right)}}}{8}$$

逐項積分：

$$\frac{15 \sin{\left(7 x \right)}}{56} + \frac{5 \sin{\left(21 x \right)}}{168} - \frac{\int{5 \cos{\left(14 x \right)} \cos{\left(21 x \right)} d x}}{8} - \frac{{\color{red}{\int{\left(15 \cos{\left(7 x \right)} + 15 \cos{\left(21 x \right)}\right)d x}}}}{16} = \frac{15 \sin{\left(7 x \right)}}{56} + \frac{5 \sin{\left(21 x \right)}}{168} - \frac{\int{5 \cos{\left(14 x \right)} \cos{\left(21 x \right)} d x}}{8} - \frac{{\color{red}{\left(\int{15 \cos{\left(7 x \right)} d x} + \int{15 \cos{\left(21 x \right)} d x}\right)}}}{16}$$

套用常數倍法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$，使用 $$$c=15$$$ 與 $$$f{\left(x \right)} = \cos{\left(7 x \right)}$$$：

$$\frac{15 \sin{\left(7 x \right)}}{56} + \frac{5 \sin{\left(21 x \right)}}{168} - \frac{\int{5 \cos{\left(14 x \right)} \cos{\left(21 x \right)} d x}}{8} - \frac{\int{15 \cos{\left(21 x \right)} d x}}{16} - \frac{{\color{red}{\int{15 \cos{\left(7 x \right)} d x}}}}{16} = \frac{15 \sin{\left(7 x \right)}}{56} + \frac{5 \sin{\left(21 x \right)}}{168} - \frac{\int{5 \cos{\left(14 x \right)} \cos{\left(21 x \right)} d x}}{8} - \frac{\int{15 \cos{\left(21 x \right)} d x}}{16} - \frac{{\color{red}{\left(15 \int{\cos{\left(7 x \right)} d x}\right)}}}{16}$$

積分 $$$\int{\cos{\left(7 x \right)} d x}$$$ 已經計算過：

$$\int{\cos{\left(7 x \right)} d x} = \frac{\sin{\left(7 x \right)}}{7}$$

因此，

$$\frac{15 \sin{\left(7 x \right)}}{56} + \frac{5 \sin{\left(21 x \right)}}{168} - \frac{\int{5 \cos{\left(14 x \right)} \cos{\left(21 x \right)} d x}}{8} - \frac{\int{15 \cos{\left(21 x \right)} d x}}{16} - \frac{15 {\color{red}{\int{\cos{\left(7 x \right)} d x}}}}{16} = \frac{15 \sin{\left(7 x \right)}}{56} + \frac{5 \sin{\left(21 x \right)}}{168} - \frac{\int{5 \cos{\left(14 x \right)} \cos{\left(21 x \right)} d x}}{8} - \frac{\int{15 \cos{\left(21 x \right)} d x}}{16} - \frac{15 {\color{red}{\left(\frac{\sin{\left(7 x \right)}}{7}\right)}}}{16}$$

套用常數倍法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$，使用 $$$c=15$$$ 與 $$$f{\left(x \right)} = \cos{\left(21 x \right)}$$$：

$$\frac{15 \sin{\left(7 x \right)}}{112} + \frac{5 \sin{\left(21 x \right)}}{168} - \frac{\int{5 \cos{\left(14 x \right)} \cos{\left(21 x \right)} d x}}{8} - \frac{{\color{red}{\int{15 \cos{\left(21 x \right)} d x}}}}{16} = \frac{15 \sin{\left(7 x \right)}}{112} + \frac{5 \sin{\left(21 x \right)}}{168} - \frac{\int{5 \cos{\left(14 x \right)} \cos{\left(21 x \right)} d x}}{8} - \frac{{\color{red}{\left(15 \int{\cos{\left(21 x \right)} d x}\right)}}}{16}$$

積分 $$$\int{\cos{\left(21 x \right)} d x}$$$ 已經計算過：

$$\int{\cos{\left(21 x \right)} d x} = \frac{\sin{\left(21 x \right)}}{21}$$

因此，

$$\frac{15 \sin{\left(7 x \right)}}{112} + \frac{5 \sin{\left(21 x \right)}}{168} - \frac{\int{5 \cos{\left(14 x \right)} \cos{\left(21 x \right)} d x}}{8} - \frac{15 {\color{red}{\int{\cos{\left(21 x \right)} d x}}}}{16} = \frac{15 \sin{\left(7 x \right)}}{112} + \frac{5 \sin{\left(21 x \right)}}{168} - \frac{\int{5 \cos{\left(14 x \right)} \cos{\left(21 x \right)} d x}}{8} - \frac{15 {\color{red}{\left(\frac{\sin{\left(21 x \right)}}{21}\right)}}}{16}$$

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 $$$\alpha=14 x$$$ 和 $$$\beta=21 x$$$ 將 $$$\cos\left(14 x \right)\cos\left(21 x \right)$$$ 改寫:

$$\frac{15 \sin{\left(7 x \right)}}{112} - \frac{5 \sin{\left(21 x \right)}}{336} - \frac{{\color{red}{\int{5 \cos{\left(14 x \right)} \cos{\left(21 x \right)} d x}}}}{8} = \frac{15 \sin{\left(7 x \right)}}{112} - \frac{5 \sin{\left(21 x \right)}}{336} - \frac{{\color{red}{\int{\left(\frac{5 \cos{\left(7 x \right)}}{2} + \frac{5 \cos{\left(35 x \right)}}{2}\right)d x}}}}{8}$$

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 $$$f{\left(x \right)} = 5 \cos{\left(7 x \right)} + 5 \cos{\left(35 x \right)}$$$：

$$\frac{15 \sin{\left(7 x \right)}}{112} - \frac{5 \sin{\left(21 x \right)}}{336} - \frac{{\color{red}{\int{\left(\frac{5 \cos{\left(7 x \right)}}{2} + \frac{5 \cos{\left(35 x \right)}}{2}\right)d x}}}}{8} = \frac{15 \sin{\left(7 x \right)}}{112} - \frac{5 \sin{\left(21 x \right)}}{336} - \frac{{\color{red}{\left(\frac{\int{\left(5 \cos{\left(7 x \right)} + 5 \cos{\left(35 x \right)}\right)d x}}{2}\right)}}}{8}$$

重寫被積函數:

$$\frac{15 \sin{\left(7 x \right)}}{112} - \frac{5 \sin{\left(21 x \right)}}{336} - \frac{{\color{red}{\int{\left(5 \cos{\left(7 x \right)} + 5 \cos{\left(35 x \right)}\right)d x}}}}{16} = \frac{15 \sin{\left(7 x \right)}}{112} - \frac{5 \sin{\left(21 x \right)}}{336} - \frac{{\color{red}{\int{5 \left(\cos{\left(7 x \right)} + \cos{\left(35 x \right)}\right) d x}}}}{16}$$

簡化被積函數:

$$\frac{15 \sin{\left(7 x \right)}}{112} - \frac{5 \sin{\left(21 x \right)}}{336} - \frac{{\color{red}{\int{5 \left(\cos{\left(7 x \right)} + \cos{\left(35 x \right)}\right) d x}}}}{16} = \frac{15 \sin{\left(7 x \right)}}{112} - \frac{5 \sin{\left(21 x \right)}}{336} - \frac{{\color{red}{\int{\left(5 \cos{\left(7 x \right)} + 5 \cos{\left(35 x \right)}\right)d x}}}}{16}$$

套用常數倍法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$，使用 $$$c=5$$$ 與 $$$f{\left(x \right)} = \cos{\left(7 x \right)} + \cos{\left(35 x \right)}$$$：

$$\frac{15 \sin{\left(7 x \right)}}{112} - \frac{5 \sin{\left(21 x \right)}}{336} - \frac{{\color{red}{\int{\left(5 \cos{\left(7 x \right)} + 5 \cos{\left(35 x \right)}\right)d x}}}}{16} = \frac{15 \sin{\left(7 x \right)}}{112} - \frac{5 \sin{\left(21 x \right)}}{336} - \frac{{\color{red}{\left(5 \int{\left(\cos{\left(7 x \right)} + \cos{\left(35 x \right)}\right)d x}\right)}}}{16}$$

令 $$$w=7 x$$$。

則 $$$dw=\left(7 x\right)^{\prime }dx = 7 dx$$$ (步驟見»)，並可得 $$$dx = \frac{dw}{7}$$$。

該積分可改寫為

$$\frac{15 \sin{\left(7 x \right)}}{112} - \frac{5 \sin{\left(21 x \right)}}{336} - \frac{5 {\color{red}{\int{\left(\cos{\left(7 x \right)} + \cos{\left(35 x \right)}\right)d x}}}}{16} = \frac{15 \sin{\left(7 x \right)}}{112} - \frac{5 \sin{\left(21 x \right)}}{336} - \frac{5 {\color{red}{\int{\left(\frac{\cos{\left(w \right)}}{7} + \frac{\cos{\left(5 w \right)}}{7}\right)d w}}}}{16}$$

套用常數倍法則 $$$\int c f{\left(w \right)}\, dw = c \int f{\left(w \right)}\, dw$$$，使用 $$$c=\frac{1}{7}$$$ 與 $$$f{\left(w \right)} = \cos{\left(w \right)} + \cos{\left(5 w \right)}$$$：

$$\frac{15 \sin{\left(7 x \right)}}{112} - \frac{5 \sin{\left(21 x \right)}}{336} - \frac{5 {\color{red}{\int{\left(\frac{\cos{\left(w \right)}}{7} + \frac{\cos{\left(5 w \right)}}{7}\right)d w}}}}{16} = \frac{15 \sin{\left(7 x \right)}}{112} - \frac{5 \sin{\left(21 x \right)}}{336} - \frac{5 {\color{red}{\left(\frac{\int{\left(\cos{\left(w \right)} + \cos{\left(5 w \right)}\right)d w}}{7}\right)}}}{16}$$

逐項積分：

$$\frac{15 \sin{\left(7 x \right)}}{112} - \frac{5 \sin{\left(21 x \right)}}{336} - \frac{5 {\color{red}{\int{\left(\cos{\left(w \right)} + \cos{\left(5 w \right)}\right)d w}}}}{112} = \frac{15 \sin{\left(7 x \right)}}{112} - \frac{5 \sin{\left(21 x \right)}}{336} - \frac{5 {\color{red}{\left(\int{\cos{\left(w \right)} d w} + \int{\cos{\left(5 w \right)} d w}\right)}}}{112}$$

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

$$\frac{15 \sin{\left(7 x \right)}}{112} - \frac{5 \sin{\left(21 x \right)}}{336} - \frac{5 \int{\cos{\left(5 w \right)} d w}}{112} - \frac{5 {\color{red}{\int{\cos{\left(w \right)} d w}}}}{112} = \frac{15 \sin{\left(7 x \right)}}{112} - \frac{5 \sin{\left(21 x \right)}}{336} - \frac{5 \int{\cos{\left(5 w \right)} d w}}{112} - \frac{5 {\color{red}{\sin{\left(w \right)}}}}{112}$$

令 $$$\theta=5 w$$$。

則 $$$d\theta=\left(5 w\right)^{\prime }dw = 5 dw$$$ (步驟見»)，並可得 $$$dw = \frac{d\theta}{5}$$$。

所以，

$$- \frac{5 \sin{\left(w \right)}}{112} + \frac{15 \sin{\left(7 x \right)}}{112} - \frac{5 \sin{\left(21 x \right)}}{336} - \frac{5 {\color{red}{\int{\cos{\left(5 w \right)} d w}}}}{112} = - \frac{5 \sin{\left(w \right)}}{112} + \frac{15 \sin{\left(7 x \right)}}{112} - \frac{5 \sin{\left(21 x \right)}}{336} - \frac{5 {\color{red}{\int{\frac{\cos{\left(\theta \right)}}{5} d \theta}}}}{112}$$

套用常數倍法則 $$$\int c f{\left(\theta \right)}\, d\theta = c \int f{\left(\theta \right)}\, d\theta$$$，使用 $$$c=\frac{1}{5}$$$ 與 $$$f{\left(\theta \right)} = \cos{\left(\theta \right)}$$$：

$$- \frac{5 \sin{\left(w \right)}}{112} + \frac{15 \sin{\left(7 x \right)}}{112} - \frac{5 \sin{\left(21 x \right)}}{336} - \frac{5 {\color{red}{\int{\frac{\cos{\left(\theta \right)}}{5} d \theta}}}}{112} = - \frac{5 \sin{\left(w \right)}}{112} + \frac{15 \sin{\left(7 x \right)}}{112} - \frac{5 \sin{\left(21 x \right)}}{336} - \frac{5 {\color{red}{\left(\frac{\int{\cos{\left(\theta \right)} d \theta}}{5}\right)}}}{112}$$

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

$$- \frac{5 \sin{\left(w \right)}}{112} + \frac{15 \sin{\left(7 x \right)}}{112} - \frac{5 \sin{\left(21 x \right)}}{336} - \frac{{\color{red}{\int{\cos{\left(\theta \right)} d \theta}}}}{112} = - \frac{5 \sin{\left(w \right)}}{112} + \frac{15 \sin{\left(7 x \right)}}{112} - \frac{5 \sin{\left(21 x \right)}}{336} - \frac{{\color{red}{\sin{\left(\theta \right)}}}}{112}$$

回顧一下 $$$\theta=5 w$$$：

$$- \frac{5 \sin{\left(w \right)}}{112} + \frac{15 \sin{\left(7 x \right)}}{112} - \frac{5 \sin{\left(21 x \right)}}{336} - \frac{\sin{\left({\color{red}{\theta}} \right)}}{112} = - \frac{5 \sin{\left(w \right)}}{112} + \frac{15 \sin{\left(7 x \right)}}{112} - \frac{5 \sin{\left(21 x \right)}}{336} - \frac{\sin{\left({\color{red}{\left(5 w\right)}} \right)}}{112}$$

回顧一下 $$$w=7 x$$$：

$$\frac{15 \sin{\left(7 x \right)}}{112} - \frac{5 \sin{\left(21 x \right)}}{336} - \frac{5 \sin{\left({\color{red}{w}} \right)}}{112} - \frac{\sin{\left(5 {\color{red}{w}} \right)}}{112} = \frac{15 \sin{\left(7 x \right)}}{112} - \frac{5 \sin{\left(21 x \right)}}{336} - \frac{5 \sin{\left({\color{red}{\left(7 x\right)}} \right)}}{112} - \frac{\sin{\left(5 {\color{red}{\left(7 x\right)}} \right)}}{112}$$

因此，

$$\int{5 \sin^{2}{\left(7 x \right)} \cos^{3}{\left(7 x \right)} d x} = \frac{5 \sin{\left(7 x \right)}}{56} - \frac{5 \sin{\left(21 x \right)}}{336} - \frac{\sin{\left(35 x \right)}}{112}$$

加上積分常數：

$$\int{5 \sin^{2}{\left(7 x \right)} \cos^{3}{\left(7 x \right)} d x} = \frac{5 \sin{\left(7 x \right)}}{56} - \frac{5 \sin{\left(21 x \right)}}{336} - \frac{\sin{\left(35 x \right)}}{112}+C$$

$$$\int 5 \sin^{2}{\left(7 x \right)} \cos^{3}{\left(7 x \right)}\, dx = \left(\frac{5 \sin{\left(7 x \right)}}{56} - \frac{5 \sin{\left(21 x \right)}}{336} - \frac{\sin{\left(35 x \right)}}{112}\right) + C$$$A