$$$\sin{\left(\frac{\pi x}{4} \right)}$$$ 的积分

求$$$\int \sin{\left(\frac{\pi x}{4} \right)}\, dx$$$。

设$$$u=\frac{\pi x}{4}$$$。

则$$$du=\left(\frac{\pi x}{4}\right)^{\prime }dx = \frac{\pi}{4} dx$$$ (步骤见»)，并有$$$dx = \frac{4 du}{\pi}$$$。

所以，

$${\color{red}{\int{\sin{\left(\frac{\pi x}{4} \right)} d x}}} = {\color{red}{\int{\frac{4 \sin{\left(u \right)}}{\pi} d u}}}$$

对 $$$c=\frac{4}{\pi}$$$ 和 $$$f{\left(u \right)} = \sin{\left(u \right)}$$$ 应用常数倍法则 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$：

$${\color{red}{\int{\frac{4 \sin{\left(u \right)}}{\pi} d u}}} = {\color{red}{\left(\frac{4 \int{\sin{\left(u \right)} d u}}{\pi}\right)}}$$

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

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

回忆一下 $$$u=\frac{\pi x}{4}$$$:

$$- \frac{4 \cos{\left({\color{red}{u}} \right)}}{\pi} = - \frac{4 \cos{\left({\color{red}{\left(\frac{\pi x}{4}\right)}} \right)}}{\pi}$$

因此，

$$\int{\sin{\left(\frac{\pi x}{4} \right)} d x} = - \frac{4 \cos{\left(\frac{\pi x}{4} \right)}}{\pi}$$

加上积分常数：

$$\int{\sin{\left(\frac{\pi x}{4} \right)} d x} = - \frac{4 \cos{\left(\frac{\pi x}{4} \right)}}{\pi}+C$$

$$$\int \sin{\left(\frac{\pi x}{4} \right)}\, dx = - \frac{4 \cos{\left(\frac{\pi x}{4} \right)}}{\pi} + C$$$A