$$$7 \cos{\left(7 x \right)}$$$ 的积分

求$$$\int 7 \cos{\left(7 x \right)}\, dx$$$。

对 $$$c=7$$$ 和 $$$f{\left(x \right)} = \cos{\left(7 x \right)}$$$ 应用常数倍法则 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$：

$${\color{red}{\int{7 \cos{\left(7 x \right)} d x}}} = {\color{red}{\left(7 \int{\cos{\left(7 x \right)} d x}\right)}}$$

设$$$u=7 x$$$。

则$$$du=\left(7 x\right)^{\prime }dx = 7 dx$$$ (步骤见»)，并有$$$dx = \frac{du}{7}$$$。

因此，

$$7 {\color{red}{\int{\cos{\left(7 x \right)} d x}}} = 7 {\color{red}{\int{\frac{\cos{\left(u \right)}}{7} d u}}}$$

对 $$$c=\frac{1}{7}$$$ 和 $$$f{\left(u \right)} = \cos{\left(u \right)}$$$ 应用常数倍法则 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$：

$$7 {\color{red}{\int{\frac{\cos{\left(u \right)}}{7} d u}}} = 7 {\color{red}{\left(\frac{\int{\cos{\left(u \right)} d u}}{7}\right)}}$$

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

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

回忆一下 $$$u=7 x$$$:

$$\sin{\left({\color{red}{u}} \right)} = \sin{\left({\color{red}{\left(7 x\right)}} \right)}$$

因此，

$$\int{7 \cos{\left(7 x \right)} d x} = \sin{\left(7 x \right)}$$

加上积分常数：

$$\int{7 \cos{\left(7 x \right)} d x} = \sin{\left(7 x \right)}+C$$

$$$\int 7 \cos{\left(7 x \right)}\, dx = \sin{\left(7 x \right)} + C$$$A