$$$\cos{\left(43 \theta \right)}$$$ 的积分

求$$$\int \cos{\left(43 \theta \right)}\, d\theta$$$。

设$$$u=43 \theta$$$。

则$$$du=\left(43 \theta\right)^{\prime }d\theta = 43 d\theta$$$ (步骤见»)，并有$$$d\theta = \frac{du}{43}$$$。

因此，

$${\color{red}{\int{\cos{\left(43 \theta \right)} d \theta}}} = {\color{red}{\int{\frac{\cos{\left(u \right)}}{43} d u}}}$$

对 $$$c=\frac{1}{43}$$$ 和 $$$f{\left(u \right)} = \cos{\left(u \right)}$$$ 应用常数倍法则 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$：

$${\color{red}{\int{\frac{\cos{\left(u \right)}}{43} d u}}} = {\color{red}{\left(\frac{\int{\cos{\left(u \right)} d u}}{43}\right)}}$$

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

$$\frac{{\color{red}{\int{\cos{\left(u \right)} d u}}}}{43} = \frac{{\color{red}{\sin{\left(u \right)}}}}{43}$$

回忆一下 $$$u=43 \theta$$$:

$$\frac{\sin{\left({\color{red}{u}} \right)}}{43} = \frac{\sin{\left({\color{red}{\left(43 \theta\right)}} \right)}}{43}$$

因此，

$$\int{\cos{\left(43 \theta \right)} d \theta} = \frac{\sin{\left(43 \theta \right)}}{43}$$

加上积分常数：

$$\int{\cos{\left(43 \theta \right)} d \theta} = \frac{\sin{\left(43 \theta \right)}}{43}+C$$

$$$\int \cos{\left(43 \theta \right)}\, d\theta = \frac{\sin{\left(43 \theta \right)}}{43} + C$$$A