$$$\frac{b^{2} \sin{\left(x \right)} \sin{\left(2 x \right)} \cos{\left(x \right)}}{a^{2}}$$$ 對 $$$x$$$ 的積分

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

重寫被積函數:

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

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

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

使用倍角公式 $$$\sin\left(x \right)\cos\left(x \right)=\frac{1}{2}\sin\left( 2 x \right)$$$ 改寫被積函數:

$$\frac{2 b^{2} {\color{red}{\int{\sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)} d x}}}}{a^{2}} = \frac{2 b^{2} {\color{red}{\int{\frac{\sin^{2}{\left(2 x \right)}}{4} d x}}}}{a^{2}}$$

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

$$\frac{2 b^{2} {\color{red}{\int{\frac{\sin^{2}{\left(2 x \right)}}{4} d x}}}}{a^{2}} = \frac{2 b^{2} {\color{red}{\left(\frac{\int{\sin^{2}{\left(2 x \right)} d x}}{4}\right)}}}{a^{2}}$$

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

$$\frac{b^{2} {\color{red}{\int{\sin^{2}{\left(2 x \right)} d x}}}}{2 a^{2}} = \frac{b^{2} {\color{red}{\int{\left(\frac{1}{2} - \frac{\cos{\left(4 x \right)}}{2}\right)d x}}}}{2 a^{2}}$$

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 $$$f{\left(x \right)} = 1 - \cos{\left(4 x \right)}$$$：

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

逐項積分：

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

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

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

令 $$$u=4 x$$$。

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

所以，

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

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

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

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

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

回顧一下 $$$u=4 x$$$：

$$\frac{b^{2} \left(x - \frac{\sin{\left({\color{red}{u}} \right)}}{4}\right)}{4 a^{2}} = \frac{b^{2} \left(x - \frac{\sin{\left({\color{red}{\left(4 x\right)}} \right)}}{4}\right)}{4 a^{2}}$$

因此，

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

化簡：

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

加上積分常數：

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

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