$$$\frac{b^{2} \sin{\left(x \right)} \sin{\left(2 x \right)} \cos{\left(x \right)}}{a^{2}}$$$ 关于$$$x$$$的积分

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

改写被积函数:

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

对 $$$c=\frac{2 b^{2}}{a^{2}}$$$ 和 $$$f{\left(x \right)} = \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}$$$ 应用常数倍法则 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$：

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

利用二倍角公式 $$$\sin\left(x \right)\cos\left(x \right)=\frac{1}{2}\sin\left( 2 x \right)$$$ 重写被积函数:

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

对 $$$c=\frac{1}{4}$$$ 和 $$$f{\left(x \right)} = \sin^{2}{\left(2 x \right)}$$$ 应用常数倍法则 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$：

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

应用降幂公式 $$$\sin^{2}{\left(\alpha \right)} = \frac{1}{2} - \frac{\cos{\left(2 \alpha \right)}}{2}$$$，并令 $$$\alpha=2 x$$$:

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

对 $$$c=\frac{1}{2}$$$ 和 $$$f{\left(x \right)} = 1 - \cos{\left(4 x \right)}$$$ 应用常数倍法则 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$：

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

逐项积分：

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

应用常数法则 $$$\int c\, dx = c x$$$，使用 $$$c=1$$$：

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

设$$$u=4 x$$$。

则$$$du=\left(4 x\right)^{\prime }dx = 4 dx$$$ (步骤见»)，并有$$$dx = \frac{du}{4}$$$。

因此，

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

对 $$$c=\frac{1}{4}$$$ 和 $$$f{\left(u \right)} = \cos{\left(u \right)}$$$ 应用常数倍法则 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$：

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

余弦函数的积分为 $$$\int{\cos{\left(u \right)} d u} = \sin{\left(u \right)}$$$：

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

回忆一下 $$$u=4 x$$$:

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

因此，

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

化简：

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

加上积分常数：

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

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