$$$2 \cos{\left(\pi t \right)}$$$ 的积分

求$$$\int 2 \cos{\left(\pi t \right)}\, dt$$$。

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

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

设$$$u=\pi t$$$。

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

该积分可以改写为

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

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

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

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

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

回忆一下 $$$u=\pi t$$$:

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

因此，

$$\int{2 \cos{\left(\pi t \right)} d t} = \frac{2 \sin{\left(\pi t \right)}}{\pi}$$

加上积分常数：

$$\int{2 \cos{\left(\pi t \right)} d t} = \frac{2 \sin{\left(\pi t \right)}}{\pi}+C$$

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