$$$5 \cos{\left(x \right)} \cos{\left(5 x \right)}$$$ 的积分

求$$$\int 5 \cos{\left(x \right)} \cos{\left(5 x \right)}\, dx$$$。

使用公式 $$$\cos\left(\alpha \right)\cos\left(\beta \right)=\frac{1}{2} \cos\left(\alpha-\beta \right)+\frac{1}{2} \cos\left(\alpha+\beta \right)$$$，取 $$$\alpha=x$$$ 和 $$$\beta=5 x$$$，将 $$$\cos\left(x \right)\cos\left(5 x \right)$$$ 重写:

$${\color{red}{\int{5 \cos{\left(x \right)} \cos{\left(5 x \right)} d x}}} = {\color{red}{\int{\left(\frac{5 \cos{\left(4 x \right)}}{2} + \frac{5 \cos{\left(6 x \right)}}{2}\right)d x}}}$$

对 $$$c=\frac{1}{2}$$$ 和 $$$f{\left(x \right)} = 5 \cos{\left(4 x \right)} + 5 \cos{\left(6 x \right)}$$$ 应用常数倍法则 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$：

$${\color{red}{\int{\left(\frac{5 \cos{\left(4 x \right)}}{2} + \frac{5 \cos{\left(6 x \right)}}{2}\right)d x}}} = {\color{red}{\left(\frac{\int{\left(5 \cos{\left(4 x \right)} + 5 \cos{\left(6 x \right)}\right)d x}}{2}\right)}}$$

逐项积分：

$$\frac{{\color{red}{\int{\left(5 \cos{\left(4 x \right)} + 5 \cos{\left(6 x \right)}\right)d x}}}}{2} = \frac{{\color{red}{\left(\int{5 \cos{\left(4 x \right)} d x} + \int{5 \cos{\left(6 x \right)} d x}\right)}}}{2}$$

对 $$$c=5$$$ 和 $$$f{\left(x \right)} = \cos{\left(4 x \right)}$$$ 应用常数倍法则 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$：

$$\frac{\int{5 \cos{\left(6 x \right)} d x}}{2} + \frac{{\color{red}{\int{5 \cos{\left(4 x \right)} d x}}}}{2} = \frac{\int{5 \cos{\left(6 x \right)} d x}}{2} + \frac{{\color{red}{\left(5 \int{\cos{\left(4 x \right)} d x}\right)}}}{2}$$

设$$$u=4 x$$$。

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

因此，

$$\frac{\int{5 \cos{\left(6 x \right)} d x}}{2} + \frac{5 {\color{red}{\int{\cos{\left(4 x \right)} d x}}}}{2} = \frac{\int{5 \cos{\left(6 x \right)} d x}}{2} + \frac{5 {\color{red}{\int{\frac{\cos{\left(u \right)}}{4} d u}}}}{2}$$

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

$$\frac{\int{5 \cos{\left(6 x \right)} d x}}{2} + \frac{5 {\color{red}{\int{\frac{\cos{\left(u \right)}}{4} d u}}}}{2} = \frac{\int{5 \cos{\left(6 x \right)} d x}}{2} + \frac{5 {\color{red}{\left(\frac{\int{\cos{\left(u \right)} d u}}{4}\right)}}}{2}$$

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

$$\frac{\int{5 \cos{\left(6 x \right)} d x}}{2} + \frac{5 {\color{red}{\int{\cos{\left(u \right)} d u}}}}{8} = \frac{\int{5 \cos{\left(6 x \right)} d x}}{2} + \frac{5 {\color{red}{\sin{\left(u \right)}}}}{8}$$

回忆一下 $$$u=4 x$$$:

$$\frac{\int{5 \cos{\left(6 x \right)} d x}}{2} + \frac{5 \sin{\left({\color{red}{u}} \right)}}{8} = \frac{\int{5 \cos{\left(6 x \right)} d x}}{2} + \frac{5 \sin{\left({\color{red}{\left(4 x\right)}} \right)}}{8}$$

对 $$$c=5$$$ 和 $$$f{\left(x \right)} = \cos{\left(6 x \right)}$$$ 应用常数倍法则 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$：

$$\frac{5 \sin{\left(4 x \right)}}{8} + \frac{{\color{red}{\int{5 \cos{\left(6 x \right)} d x}}}}{2} = \frac{5 \sin{\left(4 x \right)}}{8} + \frac{{\color{red}{\left(5 \int{\cos{\left(6 x \right)} d x}\right)}}}{2}$$

设$$$u=6 x$$$。

则$$$du=\left(6 x\right)^{\prime }dx = 6 dx$$$ (步骤见»)，并有$$$dx = \frac{du}{6}$$$。

该积分可以改写为

$$\frac{5 \sin{\left(4 x \right)}}{8} + \frac{5 {\color{red}{\int{\cos{\left(6 x \right)} d x}}}}{2} = \frac{5 \sin{\left(4 x \right)}}{8} + \frac{5 {\color{red}{\int{\frac{\cos{\left(u \right)}}{6} d u}}}}{2}$$

对 $$$c=\frac{1}{6}$$$ 和 $$$f{\left(u \right)} = \cos{\left(u \right)}$$$ 应用常数倍法则 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$：

$$\frac{5 \sin{\left(4 x \right)}}{8} + \frac{5 {\color{red}{\int{\frac{\cos{\left(u \right)}}{6} d u}}}}{2} = \frac{5 \sin{\left(4 x \right)}}{8} + \frac{5 {\color{red}{\left(\frac{\int{\cos{\left(u \right)} d u}}{6}\right)}}}{2}$$

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

$$\frac{5 \sin{\left(4 x \right)}}{8} + \frac{5 {\color{red}{\int{\cos{\left(u \right)} d u}}}}{12} = \frac{5 \sin{\left(4 x \right)}}{8} + \frac{5 {\color{red}{\sin{\left(u \right)}}}}{12}$$

回忆一下 $$$u=6 x$$$:

$$\frac{5 \sin{\left(4 x \right)}}{8} + \frac{5 \sin{\left({\color{red}{u}} \right)}}{12} = \frac{5 \sin{\left(4 x \right)}}{8} + \frac{5 \sin{\left({\color{red}{\left(6 x\right)}} \right)}}{12}$$

因此，

$$\int{5 \cos{\left(x \right)} \cos{\left(5 x \right)} d x} = \frac{5 \sin{\left(4 x \right)}}{8} + \frac{5 \sin{\left(6 x \right)}}{12}$$

化简：

$$\int{5 \cos{\left(x \right)} \cos{\left(5 x \right)} d x} = \frac{5 \left(3 \sin{\left(4 x \right)} + 2 \sin{\left(6 x \right)}\right)}{24}$$

加上积分常数：

$$\int{5 \cos{\left(x \right)} \cos{\left(5 x \right)} d x} = \frac{5 \left(3 \sin{\left(4 x \right)} + 2 \sin{\left(6 x \right)}\right)}{24}+C$$

$$$\int 5 \cos{\left(x \right)} \cos{\left(5 x \right)}\, dx = \frac{5 \left(3 \sin{\left(4 x \right)} + 2 \sin{\left(6 x \right)}\right)}{24} + C$$$A