$$$\left(2 - 3 \sin{\left(x \right)}\right) \cos^{2}{\left(x \right)}$$$ 的积分

求$$$\int \left(2 - 3 \sin{\left(x \right)}\right) \cos^{2}{\left(x \right)}\, dx$$$。

应用降幂公式 $$$\cos^{2}{\left(\alpha \right)} = \frac{\cos{\left(2 \alpha \right)}}{2} + \frac{1}{2}$$$，并令 $$$\alpha=x$$$:

$${\color{red}{\int{\left(2 - 3 \sin{\left(x \right)}\right) \cos^{2}{\left(x \right)} d x}}} = {\color{red}{\int{\frac{\left(2 - 3 \sin{\left(x \right)}\right) \left(\cos{\left(2 x \right)} + 1\right)}{2} d x}}}$$

对 $$$c=\frac{1}{2}$$$ 和 $$$f{\left(x \right)} = \left(2 - 3 \sin{\left(x \right)}\right) \left(\cos{\left(2 x \right)} + 1\right)$$$ 应用常数倍法则 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$：

$${\color{red}{\int{\frac{\left(2 - 3 \sin{\left(x \right)}\right) \left(\cos{\left(2 x \right)} + 1\right)}{2} d x}}} = {\color{red}{\left(\frac{\int{\left(2 - 3 \sin{\left(x \right)}\right) \left(\cos{\left(2 x \right)} + 1\right) d x}}{2}\right)}}$$

Expand the expression:

$$\frac{{\color{red}{\int{\left(2 - 3 \sin{\left(x \right)}\right) \left(\cos{\left(2 x \right)} + 1\right) d x}}}}{2} = \frac{{\color{red}{\int{\left(- 3 \sin{\left(x \right)} \cos{\left(2 x \right)} - 3 \sin{\left(x \right)} + 2 \cos{\left(2 x \right)} + 2\right)d x}}}}{2}$$

逐项积分：

$$\frac{{\color{red}{\int{\left(- 3 \sin{\left(x \right)} \cos{\left(2 x \right)} - 3 \sin{\left(x \right)} + 2 \cos{\left(2 x \right)} + 2\right)d x}}}}{2} = \frac{{\color{red}{\left(\int{2 d x} - \int{3 \sin{\left(x \right)} \cos{\left(2 x \right)} d x} - \int{3 \sin{\left(x \right)} d x} + \int{2 \cos{\left(2 x \right)} d x}\right)}}}{2}$$

应用常数法则 $$$\int c\, dx = c x$$$，使用 $$$c=2$$$：

$$- \frac{\int{3 \sin{\left(x \right)} \cos{\left(2 x \right)} d x}}{2} - \frac{\int{3 \sin{\left(x \right)} d x}}{2} + \frac{\int{2 \cos{\left(2 x \right)} d x}}{2} + \frac{{\color{red}{\int{2 d x}}}}{2} = - \frac{\int{3 \sin{\left(x \right)} \cos{\left(2 x \right)} d x}}{2} - \frac{\int{3 \sin{\left(x \right)} d x}}{2} + \frac{\int{2 \cos{\left(2 x \right)} d x}}{2} + \frac{{\color{red}{\left(2 x\right)}}}{2}$$

对 $$$c=3$$$ 和 $$$f{\left(x \right)} = \sin{\left(x \right)}$$$ 应用常数倍法则 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$：

$$x - \frac{\int{3 \sin{\left(x \right)} \cos{\left(2 x \right)} d x}}{2} + \frac{\int{2 \cos{\left(2 x \right)} d x}}{2} - \frac{{\color{red}{\int{3 \sin{\left(x \right)} d x}}}}{2} = x - \frac{\int{3 \sin{\left(x \right)} \cos{\left(2 x \right)} d x}}{2} + \frac{\int{2 \cos{\left(2 x \right)} d x}}{2} - \frac{{\color{red}{\left(3 \int{\sin{\left(x \right)} d x}\right)}}}{2}$$

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

$$x - \frac{\int{3 \sin{\left(x \right)} \cos{\left(2 x \right)} d x}}{2} + \frac{\int{2 \cos{\left(2 x \right)} d x}}{2} - \frac{3 {\color{red}{\int{\sin{\left(x \right)} d x}}}}{2} = x - \frac{\int{3 \sin{\left(x \right)} \cos{\left(2 x \right)} d x}}{2} + \frac{\int{2 \cos{\left(2 x \right)} d x}}{2} - \frac{3 {\color{red}{\left(- \cos{\left(x \right)}\right)}}}{2}$$

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

$$x + \frac{3 \cos{\left(x \right)}}{2} - \frac{\int{3 \sin{\left(x \right)} \cos{\left(2 x \right)} d x}}{2} + \frac{{\color{red}{\int{2 \cos{\left(2 x \right)} d x}}}}{2} = x + \frac{3 \cos{\left(x \right)}}{2} - \frac{\int{3 \sin{\left(x \right)} \cos{\left(2 x \right)} d x}}{2} + \frac{{\color{red}{\left(2 \int{\cos{\left(2 x \right)} d x}\right)}}}{2}$$

设$$$u=2 x$$$。

则$$$du=\left(2 x\right)^{\prime }dx = 2 dx$$$ (步骤见»)，并有$$$dx = \frac{du}{2}$$$。

所以，

$$x + \frac{3 \cos{\left(x \right)}}{2} - \frac{\int{3 \sin{\left(x \right)} \cos{\left(2 x \right)} d x}}{2} + {\color{red}{\int{\cos{\left(2 x \right)} d x}}} = x + \frac{3 \cos{\left(x \right)}}{2} - \frac{\int{3 \sin{\left(x \right)} \cos{\left(2 x \right)} d x}}{2} + {\color{red}{\int{\frac{\cos{\left(u \right)}}{2} d u}}}$$

对 $$$c=\frac{1}{2}$$$ 和 $$$f{\left(u \right)} = \cos{\left(u \right)}$$$ 应用常数倍法则 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$：

$$x + \frac{3 \cos{\left(x \right)}}{2} - \frac{\int{3 \sin{\left(x \right)} \cos{\left(2 x \right)} d x}}{2} + {\color{red}{\int{\frac{\cos{\left(u \right)}}{2} d u}}} = x + \frac{3 \cos{\left(x \right)}}{2} - \frac{\int{3 \sin{\left(x \right)} \cos{\left(2 x \right)} d x}}{2} + {\color{red}{\left(\frac{\int{\cos{\left(u \right)} d u}}{2}\right)}}$$

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

$$x + \frac{3 \cos{\left(x \right)}}{2} - \frac{\int{3 \sin{\left(x \right)} \cos{\left(2 x \right)} d x}}{2} + \frac{{\color{red}{\int{\cos{\left(u \right)} d u}}}}{2} = x + \frac{3 \cos{\left(x \right)}}{2} - \frac{\int{3 \sin{\left(x \right)} \cos{\left(2 x \right)} d x}}{2} + \frac{{\color{red}{\sin{\left(u \right)}}}}{2}$$

回忆一下 $$$u=2 x$$$:

$$x + \frac{3 \cos{\left(x \right)}}{2} - \frac{\int{3 \sin{\left(x \right)} \cos{\left(2 x \right)} d x}}{2} + \frac{\sin{\left({\color{red}{u}} \right)}}{2} = x + \frac{3 \cos{\left(x \right)}}{2} - \frac{\int{3 \sin{\left(x \right)} \cos{\left(2 x \right)} d x}}{2} + \frac{\sin{\left({\color{red}{\left(2 x\right)}} \right)}}{2}$$

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

$$x + \frac{\sin{\left(2 x \right)}}{2} + \frac{3 \cos{\left(x \right)}}{2} - \frac{{\color{red}{\int{3 \sin{\left(x \right)} \cos{\left(2 x \right)} d x}}}}{2} = x + \frac{\sin{\left(2 x \right)}}{2} + \frac{3 \cos{\left(x \right)}}{2} - \frac{{\color{red}{\int{\left(- \frac{3 \sin{\left(x \right)}}{2} + \frac{3 \sin{\left(3 x \right)}}{2}\right)d x}}}}{2}$$

对 $$$c=\frac{1}{2}$$$ 和 $$$f{\left(x \right)} = - 3 \sin{\left(x \right)} + 3 \sin{\left(3 x \right)}$$$ 应用常数倍法则 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$：

$$x + \frac{\sin{\left(2 x \right)}}{2} + \frac{3 \cos{\left(x \right)}}{2} - \frac{{\color{red}{\int{\left(- \frac{3 \sin{\left(x \right)}}{2} + \frac{3 \sin{\left(3 x \right)}}{2}\right)d x}}}}{2} = x + \frac{\sin{\left(2 x \right)}}{2} + \frac{3 \cos{\left(x \right)}}{2} - \frac{{\color{red}{\left(\frac{\int{\left(- 3 \sin{\left(x \right)} + 3 \sin{\left(3 x \right)}\right)d x}}{2}\right)}}}{2}$$

逐项积分：

$$x + \frac{\sin{\left(2 x \right)}}{2} + \frac{3 \cos{\left(x \right)}}{2} - \frac{{\color{red}{\int{\left(- 3 \sin{\left(x \right)} + 3 \sin{\left(3 x \right)}\right)d x}}}}{4} = x + \frac{\sin{\left(2 x \right)}}{2} + \frac{3 \cos{\left(x \right)}}{2} - \frac{{\color{red}{\left(- \int{3 \sin{\left(x \right)} d x} + \int{3 \sin{\left(3 x \right)} d x}\right)}}}{4}$$

对 $$$c=3$$$ 和 $$$f{\left(x \right)} = \sin{\left(x \right)}$$$ 应用常数倍法则 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$：

$$x + \frac{\sin{\left(2 x \right)}}{2} + \frac{3 \cos{\left(x \right)}}{2} - \frac{\int{3 \sin{\left(3 x \right)} d x}}{4} + \frac{{\color{red}{\int{3 \sin{\left(x \right)} d x}}}}{4} = x + \frac{\sin{\left(2 x \right)}}{2} + \frac{3 \cos{\left(x \right)}}{2} - \frac{\int{3 \sin{\left(3 x \right)} d x}}{4} + \frac{{\color{red}{\left(3 \int{\sin{\left(x \right)} d x}\right)}}}{4}$$

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

$$x + \frac{\sin{\left(2 x \right)}}{2} + \frac{3 \cos{\left(x \right)}}{2} - \frac{\int{3 \sin{\left(3 x \right)} d x}}{4} + \frac{3 {\color{red}{\int{\sin{\left(x \right)} d x}}}}{4} = x + \frac{\sin{\left(2 x \right)}}{2} + \frac{3 \cos{\left(x \right)}}{2} - \frac{\int{3 \sin{\left(3 x \right)} d x}}{4} + \frac{3 {\color{red}{\left(- \cos{\left(x \right)}\right)}}}{4}$$

对 $$$c=3$$$ 和 $$$f{\left(x \right)} = \sin{\left(3 x \right)}$$$ 应用常数倍法则 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$：

$$x + \frac{\sin{\left(2 x \right)}}{2} + \frac{3 \cos{\left(x \right)}}{4} - \frac{{\color{red}{\int{3 \sin{\left(3 x \right)} d x}}}}{4} = x + \frac{\sin{\left(2 x \right)}}{2} + \frac{3 \cos{\left(x \right)}}{4} - \frac{{\color{red}{\left(3 \int{\sin{\left(3 x \right)} d x}\right)}}}{4}$$

设$$$u=3 x$$$。

则$$$du=\left(3 x\right)^{\prime }dx = 3 dx$$$ (步骤见»)，并有$$$dx = \frac{du}{3}$$$。

积分变为

$$x + \frac{\sin{\left(2 x \right)}}{2} + \frac{3 \cos{\left(x \right)}}{4} - \frac{3 {\color{red}{\int{\sin{\left(3 x \right)} d x}}}}{4} = x + \frac{\sin{\left(2 x \right)}}{2} + \frac{3 \cos{\left(x \right)}}{4} - \frac{3 {\color{red}{\int{\frac{\sin{\left(u \right)}}{3} d u}}}}{4}$$

对 $$$c=\frac{1}{3}$$$ 和 $$$f{\left(u \right)} = \sin{\left(u \right)}$$$ 应用常数倍法则 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$：

$$x + \frac{\sin{\left(2 x \right)}}{2} + \frac{3 \cos{\left(x \right)}}{4} - \frac{3 {\color{red}{\int{\frac{\sin{\left(u \right)}}{3} d u}}}}{4} = x + \frac{\sin{\left(2 x \right)}}{2} + \frac{3 \cos{\left(x \right)}}{4} - \frac{3 {\color{red}{\left(\frac{\int{\sin{\left(u \right)} d u}}{3}\right)}}}{4}$$

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

$$x + \frac{\sin{\left(2 x \right)}}{2} + \frac{3 \cos{\left(x \right)}}{4} - \frac{{\color{red}{\int{\sin{\left(u \right)} d u}}}}{4} = x + \frac{\sin{\left(2 x \right)}}{2} + \frac{3 \cos{\left(x \right)}}{4} - \frac{{\color{red}{\left(- \cos{\left(u \right)}\right)}}}{4}$$

回忆一下 $$$u=3 x$$$:

$$x + \frac{\sin{\left(2 x \right)}}{2} + \frac{3 \cos{\left(x \right)}}{4} + \frac{\cos{\left({\color{red}{u}} \right)}}{4} = x + \frac{\sin{\left(2 x \right)}}{2} + \frac{3 \cos{\left(x \right)}}{4} + \frac{\cos{\left({\color{red}{\left(3 x\right)}} \right)}}{4}$$

因此，

$$\int{\left(2 - 3 \sin{\left(x \right)}\right) \cos^{2}{\left(x \right)} d x} = x + \frac{\sin{\left(2 x \right)}}{2} + \frac{3 \cos{\left(x \right)}}{4} + \frac{\cos{\left(3 x \right)}}{4}$$

加上积分常数：

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

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