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

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

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

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

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

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

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

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

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

$$\frac{{\color{red}{\int{\left(\frac{1}{2} - \frac{\cos{\left(4 \theta \right)}}{2}\right)d \theta}}}}{4} = \frac{{\color{red}{\left(\frac{\int{\left(1 - \cos{\left(4 \theta \right)}\right)d \theta}}{2}\right)}}}{4}$$

逐项积分：

$$\frac{{\color{red}{\int{\left(1 - \cos{\left(4 \theta \right)}\right)d \theta}}}}{8} = \frac{{\color{red}{\left(\int{1 d \theta} - \int{\cos{\left(4 \theta \right)} d \theta}\right)}}}{8}$$

应用常数法则 $$$\int c\, d\theta = c \theta$$$，使用 $$$c=1$$$：

$$- \frac{\int{\cos{\left(4 \theta \right)} d \theta}}{8} + \frac{{\color{red}{\int{1 d \theta}}}}{8} = - \frac{\int{\cos{\left(4 \theta \right)} d \theta}}{8} + \frac{{\color{red}{\theta}}}{8}$$

设$$$u=4 \theta$$$。

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

因此，

$$\frac{\theta}{8} - \frac{{\color{red}{\int{\cos{\left(4 \theta \right)} d \theta}}}}{8} = \frac{\theta}{8} - \frac{{\color{red}{\int{\frac{\cos{\left(u \right)}}{4} d u}}}}{8}$$

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

$$\frac{\theta}{8} - \frac{{\color{red}{\int{\frac{\cos{\left(u \right)}}{4} d u}}}}{8} = \frac{\theta}{8} - \frac{{\color{red}{\left(\frac{\int{\cos{\left(u \right)} d u}}{4}\right)}}}{8}$$

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

$$\frac{\theta}{8} - \frac{{\color{red}{\int{\cos{\left(u \right)} d u}}}}{32} = \frac{\theta}{8} - \frac{{\color{red}{\sin{\left(u \right)}}}}{32}$$

回忆一下 $$$u=4 \theta$$$:

$$\frac{\theta}{8} - \frac{\sin{\left({\color{red}{u}} \right)}}{32} = \frac{\theta}{8} - \frac{\sin{\left({\color{red}{\left(4 \theta\right)}} \right)}}{32}$$

因此，

$$\int{\sin^{2}{\left(\theta \right)} \cos^{2}{\left(\theta \right)} d \theta} = \frac{\theta}{8} - \frac{\sin{\left(4 \theta \right)}}{32}$$

加上积分常数：

$$\int{\sin^{2}{\left(\theta \right)} \cos^{2}{\left(\theta \right)} d \theta} = \frac{\theta}{8} - \frac{\sin{\left(4 \theta \right)}}{32}+C$$

$$$\int \sin^{2}{\left(\theta \right)} \cos^{2}{\left(\theta \right)}\, d\theta = \left(\frac{\theta}{8} - \frac{\sin{\left(4 \theta \right)}}{32}\right) + C$$$A