$$$4 \pi \sin{\left(\pi x \right)}$$$ 的积分
您的输入
求$$$\int 4 \pi \sin{\left(\pi x \right)}\, dx$$$。
解答
对 $$$c=4 \pi$$$ 和 $$$f{\left(x \right)} = \sin{\left(\pi x \right)}$$$ 应用常数倍法则 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$：
$${\color{red}{\int{4 \pi \sin{\left(\pi x \right)} d x}}} = {\color{red}{\left(4 \pi \int{\sin{\left(\pi x \right)} d x}\right)}}$$
设$$$u=\pi x$$$。
则$$$du=\left(\pi x\right)^{\prime }dx = \pi dx$$$ (步骤见»)，并有$$$dx = \frac{du}{\pi}$$$。
所以，
$$4 \pi {\color{red}{\int{\sin{\left(\pi x \right)} d x}}} = 4 \pi {\color{red}{\int{\frac{\sin{\left(u \right)}}{\pi} d u}}}$$
对 $$$c=\frac{1}{\pi}$$$ 和 $$$f{\left(u \right)} = \sin{\left(u \right)}$$$ 应用常数倍法则 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$：
$$4 \pi {\color{red}{\int{\frac{\sin{\left(u \right)}}{\pi} d u}}} = 4 \pi {\color{red}{\frac{\int{\sin{\left(u \right)} d u}}{\pi}}}$$
正弦函数的积分为 $$$\int{\sin{\left(u \right)} d u} = - \cos{\left(u \right)}$$$:
$$4 {\color{red}{\int{\sin{\left(u \right)} d u}}} = 4 {\color{red}{\left(- \cos{\left(u \right)}\right)}}$$
回忆一下 $$$u=\pi x$$$:
$$- 4 \cos{\left({\color{red}{u}} \right)} = - 4 \cos{\left({\color{red}{\pi x}} \right)}$$
因此，
$$\int{4 \pi \sin{\left(\pi x \right)} d x} = - 4 \cos{\left(\pi x \right)}$$
加上积分常数：
$$\int{4 \pi \sin{\left(\pi x \right)} d x} = - 4 \cos{\left(\pi x \right)}+C$$
答案
$$$\int 4 \pi \sin{\left(\pi x \right)}\, dx = - 4 \cos{\left(\pi x \right)} + C$$$A