$$$- \sin^{2}{\left(2 t \right)}$$$ 的积分

求$$$\int \left(- \sin^{2}{\left(2 t \right)}\right)\, dt$$$。

对 $$$c=-1$$$ 和 $$$f{\left(t \right)} = \sin^{2}{\left(2 t \right)}$$$ 应用常数倍法则 $$$\int c f{\left(t \right)}\, dt = c \int f{\left(t \right)}\, dt$$$：

$${\color{red}{\int{\left(- \sin^{2}{\left(2 t \right)}\right)d t}}} = {\color{red}{\left(- \int{\sin^{2}{\left(2 t \right)} d t}\right)}}$$

设$$$u=2 t$$$。

则$$$du=\left(2 t\right)^{\prime }dt = 2 dt$$$ (步骤见»)，并有$$$dt = \frac{du}{2}$$$。

因此，

$$- {\color{red}{\int{\sin^{2}{\left(2 t \right)} d t}}} = - {\color{red}{\int{\frac{\sin^{2}{\left(u \right)}}{2} d u}}}$$

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

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

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

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

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

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

逐项积分：

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

应用常数法则 $$$\int c\, du = c u$$$，使用 $$$c=1$$$：

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

设$$$v=2 u$$$。

则$$$dv=\left(2 u\right)^{\prime }du = 2 du$$$ (步骤见»)，并有$$$du = \frac{dv}{2}$$$。

积分变为

$$- \frac{u}{4} + \frac{{\color{red}{\int{\cos{\left(2 u \right)} d u}}}}{4} = - \frac{u}{4} + \frac{{\color{red}{\int{\frac{\cos{\left(v \right)}}{2} d v}}}}{4}$$

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

$$- \frac{u}{4} + \frac{{\color{red}{\int{\frac{\cos{\left(v \right)}}{2} d v}}}}{4} = - \frac{u}{4} + \frac{{\color{red}{\left(\frac{\int{\cos{\left(v \right)} d v}}{2}\right)}}}{4}$$

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

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

回忆一下 $$$v=2 u$$$:

$$- \frac{u}{4} + \frac{\sin{\left({\color{red}{v}} \right)}}{8} = - \frac{u}{4} + \frac{\sin{\left({\color{red}{\left(2 u\right)}} \right)}}{8}$$

回忆一下 $$$u=2 t$$$:

$$\frac{\sin{\left(2 {\color{red}{u}} \right)}}{8} - \frac{{\color{red}{u}}}{4} = \frac{\sin{\left(2 {\color{red}{\left(2 t\right)}} \right)}}{8} - \frac{{\color{red}{\left(2 t\right)}}}{4}$$

因此，

$$\int{\left(- \sin^{2}{\left(2 t \right)}\right)d t} = - \frac{t}{2} + \frac{\sin{\left(4 t \right)}}{8}$$

加上积分常数：

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

$$$\int \left(- \sin^{2}{\left(2 t \right)}\right)\, dt = \left(- \frac{t}{2} + \frac{\sin{\left(4 t \right)}}{8}\right) + C$$$A