$$$\sin{\left(2 \theta \right)} \cos{\left(\theta \right)}$$$ 的积分

求$$$\int \sin{\left(2 \theta \right)} \cos{\left(\theta \right)}\, d\theta$$$。

使用公式 $$$\sin\left(\alpha \right)\cos\left(\beta \right)=\frac{1}{2} \sin\left(\alpha-\beta \right)+\frac{1}{2} \sin\left(\alpha+\beta \right)$$$ 并结合 $$$\alpha=2 \theta$$$ 和 $$$\beta=\theta$$$ 重写被积函数:

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

对 $$$c=\frac{1}{2}$$$ 和 $$$f{\left(\theta \right)} = \sin{\left(\theta \right)} + \sin{\left(3 \theta \right)}$$$ 应用常数倍法则 $$$\int c f{\left(\theta \right)}\, d\theta = c \int f{\left(\theta \right)}\, d\theta$$$：

$${\color{red}{\int{\left(\frac{\sin{\left(\theta \right)}}{2} + \frac{\sin{\left(3 \theta \right)}}{2}\right)d \theta}}} = {\color{red}{\left(\frac{\int{\left(\sin{\left(\theta \right)} + \sin{\left(3 \theta \right)}\right)d \theta}}{2}\right)}}$$

逐项积分：

$$\frac{{\color{red}{\int{\left(\sin{\left(\theta \right)} + \sin{\left(3 \theta \right)}\right)d \theta}}}}{2} = \frac{{\color{red}{\left(\int{\sin{\left(\theta \right)} d \theta} + \int{\sin{\left(3 \theta \right)} d \theta}\right)}}}{2}$$

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

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

设$$$u=3 \theta$$$。

则$$$du=\left(3 \theta\right)^{\prime }d\theta = 3 d\theta$$$ (步骤见»)，并有$$$d\theta = \frac{du}{3}$$$。

所以，

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

对 $$$c=\frac{1}{3}$$$ 和 $$$f{\left(u \right)} = \sin{\left(u \right)}$$$ 应用常数倍法则 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$：

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

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

$$- \frac{\cos{\left(\theta \right)}}{2} + \frac{{\color{red}{\int{\sin{\left(u \right)} d u}}}}{6} = - \frac{\cos{\left(\theta \right)}}{2} + \frac{{\color{red}{\left(- \cos{\left(u \right)}\right)}}}{6}$$

回忆一下 $$$u=3 \theta$$$:

$$- \frac{\cos{\left(\theta \right)}}{2} - \frac{\cos{\left({\color{red}{u}} \right)}}{6} = - \frac{\cos{\left(\theta \right)}}{2} - \frac{\cos{\left({\color{red}{\left(3 \theta\right)}} \right)}}{6}$$

因此，

$$\int{\sin{\left(2 \theta \right)} \cos{\left(\theta \right)} d \theta} = - \frac{\cos{\left(\theta \right)}}{2} - \frac{\cos{\left(3 \theta \right)}}{6}$$

加上积分常数：

$$\int{\sin{\left(2 \theta \right)} \cos{\left(\theta \right)} d \theta} = - \frac{\cos{\left(\theta \right)}}{2} - \frac{\cos{\left(3 \theta \right)}}{6}+C$$

$$$\int \sin{\left(2 \theta \right)} \cos{\left(\theta \right)}\, d\theta = \left(- \frac{\cos{\left(\theta \right)}}{2} - \frac{\cos{\left(3 \theta \right)}}{6}\right) + C$$$A