$$$\frac{1}{2} - \frac{\cos{\left(6 t \right)}}{2}$$$ 的积分

求$$$\int \left(\frac{1}{2} - \frac{\cos{\left(6 t \right)}}{2}\right)\, dt$$$。

逐项积分：

$${\color{red}{\int{\left(\frac{1}{2} - \frac{\cos{\left(6 t \right)}}{2}\right)d t}}} = {\color{red}{\left(\int{\frac{1}{2} d t} - \int{\frac{\cos{\left(6 t \right)}}{2} d t}\right)}}$$

应用常数法则 $$$\int c\, dt = c t$$$，使用 $$$c=\frac{1}{2}$$$：

$$- \int{\frac{\cos{\left(6 t \right)}}{2} d t} + {\color{red}{\int{\frac{1}{2} d t}}} = - \int{\frac{\cos{\left(6 t \right)}}{2} d t} + {\color{red}{\left(\frac{t}{2}\right)}}$$

对 $$$c=\frac{1}{2}$$$ 和 $$$f{\left(t \right)} = \cos{\left(6 t \right)}$$$ 应用常数倍法则 $$$\int c f{\left(t \right)}\, dt = c \int f{\left(t \right)}\, dt$$$：

$$\frac{t}{2} - {\color{red}{\int{\frac{\cos{\left(6 t \right)}}{2} d t}}} = \frac{t}{2} - {\color{red}{\left(\frac{\int{\cos{\left(6 t \right)} d t}}{2}\right)}}$$

设$$$u=6 t$$$。

则$$$du=\left(6 t\right)^{\prime }dt = 6 dt$$$ (步骤见»)，并有$$$dt = \frac{du}{6}$$$。

该积分可以改写为

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

对 $$$c=\frac{1}{6}$$$ 和 $$$f{\left(u \right)} = \cos{\left(u \right)}$$$ 应用常数倍法则 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$：

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

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

$$\frac{t}{2} - \frac{{\color{red}{\int{\cos{\left(u \right)} d u}}}}{12} = \frac{t}{2} - \frac{{\color{red}{\sin{\left(u \right)}}}}{12}$$

回忆一下 $$$u=6 t$$$:

$$\frac{t}{2} - \frac{\sin{\left({\color{red}{u}} \right)}}{12} = \frac{t}{2} - \frac{\sin{\left({\color{red}{\left(6 t\right)}} \right)}}{12}$$

因此，

$$\int{\left(\frac{1}{2} - \frac{\cos{\left(6 t \right)}}{2}\right)d t} = \frac{t}{2} - \frac{\sin{\left(6 t \right)}}{12}$$

加上积分常数：

$$\int{\left(\frac{1}{2} - \frac{\cos{\left(6 t \right)}}{2}\right)d t} = \frac{t}{2} - \frac{\sin{\left(6 t \right)}}{12}+C$$

$$$\int \left(\frac{1}{2} - \frac{\cos{\left(6 t \right)}}{2}\right)\, dt = \left(\frac{t}{2} - \frac{\sin{\left(6 t \right)}}{12}\right) + C$$$A