$$$\cos{\left(\pi x \right)}$$$ 的积分

求$$$\int \cos{\left(\pi x \right)}\, dx$$$。

设$$$u=\pi x$$$。

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

因此，

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

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

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

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

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

回忆一下 $$$u=\pi x$$$:

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

因此，

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

加上积分常数：

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

$$$\int \cos{\left(\pi x \right)}\, dx = \frac{\sin{\left(\pi x \right)}}{\pi} + C$$$A