$$$\sin{\left(4 y_{} \right)}$$$ 的积分

求$$$\int \sin{\left(4 y_{} \right)}\, dy_{}$$$。

设$$$u=4 y_{}$$$。

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

该积分可以改写为

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

对 $$$c=\frac{1}{4}$$$ 和 $$$f{\left(u \right)} = \sin{\left(u \right)}$$$ 应用常数倍法则 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$：

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

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

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

回忆一下 $$$u=4 y_{}$$$:

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

因此，

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

加上积分常数：

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

$$$\int \sin{\left(4 y_{} \right)}\, dy_{} = - \frac{\cos{\left(4 y_{} \right)}}{4} + C$$$A