$$$4 \sin^{2}{\left(x \right)} \cos^{3}{\left(x \right)}$$$ 的积分

求$$$\int 4 \sin^{2}{\left(x \right)} \cos^{3}{\left(x \right)}\, dx$$$。

应用降幂公式 $$$\cos^{3}{\left(\alpha \right)} = \frac{3 \cos{\left(\alpha \right)}}{4} + \frac{\cos{\left(3 \alpha \right)}}{4}$$$，并令 $$$\alpha=x$$$:

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

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

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

对 $$$c=\frac{1}{8}$$$ 和 $$$f{\left(x \right)} = 4 \left(1 - \cos{\left(2 x \right)}\right) \left(3 \cos{\left(x \right)} + \cos{\left(3 x \right)}\right)$$$ 应用常数倍法则 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$：

$${\color{red}{\int{\frac{\left(1 - \cos{\left(2 x \right)}\right) \left(3 \cos{\left(x \right)} + \cos{\left(3 x \right)}\right)}{2} d x}}} = {\color{red}{\left(\frac{\int{4 \left(1 - \cos{\left(2 x \right)}\right) \left(3 \cos{\left(x \right)} + \cos{\left(3 x \right)}\right) d x}}{8}\right)}}$$

Expand the expression:

$$\frac{{\color{red}{\int{4 \left(1 - \cos{\left(2 x \right)}\right) \left(3 \cos{\left(x \right)} + \cos{\left(3 x \right)}\right) d x}}}}{8} = \frac{{\color{red}{\int{\left(- 12 \cos{\left(x \right)} \cos{\left(2 x \right)} + 12 \cos{\left(x \right)} - 4 \cos{\left(2 x \right)} \cos{\left(3 x \right)} + 4 \cos{\left(3 x \right)}\right)d x}}}}{8}$$

逐项积分：

$$\frac{{\color{red}{\int{\left(- 12 \cos{\left(x \right)} \cos{\left(2 x \right)} + 12 \cos{\left(x \right)} - 4 \cos{\left(2 x \right)} \cos{\left(3 x \right)} + 4 \cos{\left(3 x \right)}\right)d x}}}}{8} = \frac{{\color{red}{\left(- \int{12 \cos{\left(x \right)} \cos{\left(2 x \right)} d x} - \int{4 \cos{\left(2 x \right)} \cos{\left(3 x \right)} d x} + \int{12 \cos{\left(x \right)} d x} + \int{4 \cos{\left(3 x \right)} d x}\right)}}}{8}$$

对 $$$c=4$$$ 和 $$$f{\left(x \right)} = \cos{\left(3 x \right)}$$$ 应用常数倍法则 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$：

$$- \frac{\int{12 \cos{\left(x \right)} \cos{\left(2 x \right)} d x}}{8} - \frac{\int{4 \cos{\left(2 x \right)} \cos{\left(3 x \right)} d x}}{8} + \frac{\int{12 \cos{\left(x \right)} d x}}{8} + \frac{{\color{red}{\int{4 \cos{\left(3 x \right)} d x}}}}{8} = - \frac{\int{12 \cos{\left(x \right)} \cos{\left(2 x \right)} d x}}{8} - \frac{\int{4 \cos{\left(2 x \right)} \cos{\left(3 x \right)} d x}}{8} + \frac{\int{12 \cos{\left(x \right)} d x}}{8} + \frac{{\color{red}{\left(4 \int{\cos{\left(3 x \right)} d x}\right)}}}{8}$$

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

$$- \frac{\int{12 \cos{\left(x \right)} \cos{\left(2 x \right)} d x}}{8} - \frac{\int{4 \cos{\left(2 x \right)} \cos{\left(3 x \right)} d x}}{8} + \frac{\int{12 \cos{\left(x \right)} d x}}{8} + \frac{{\color{red}{\int{\cos{\left(3 x \right)} d x}}}}{2} = - \frac{\int{12 \cos{\left(x \right)} \cos{\left(2 x \right)} d x}}{8} - \frac{\int{4 \cos{\left(2 x \right)} \cos{\left(3 x \right)} d x}}{8} + \frac{\int{12 \cos{\left(x \right)} d x}}{8} + \frac{{\color{red}{\int{\frac{\cos{\left(u \right)}}{3} d u}}}}{2}$$

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

$$- \frac{\int{12 \cos{\left(x \right)} \cos{\left(2 x \right)} d x}}{8} - \frac{\int{4 \cos{\left(2 x \right)} \cos{\left(3 x \right)} d x}}{8} + \frac{\int{12 \cos{\left(x \right)} d x}}{8} + \frac{{\color{red}{\int{\frac{\cos{\left(u \right)}}{3} d u}}}}{2} = - \frac{\int{12 \cos{\left(x \right)} \cos{\left(2 x \right)} d x}}{8} - \frac{\int{4 \cos{\left(2 x \right)} \cos{\left(3 x \right)} d x}}{8} + \frac{\int{12 \cos{\left(x \right)} d x}}{8} + \frac{{\color{red}{\left(\frac{\int{\cos{\left(u \right)} d u}}{3}\right)}}}{2}$$

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

$$- \frac{\int{12 \cos{\left(x \right)} \cos{\left(2 x \right)} d x}}{8} - \frac{\int{4 \cos{\left(2 x \right)} \cos{\left(3 x \right)} d x}}{8} + \frac{\int{12 \cos{\left(x \right)} d x}}{8} + \frac{{\color{red}{\int{\cos{\left(u \right)} d u}}}}{6} = - \frac{\int{12 \cos{\left(x \right)} \cos{\left(2 x \right)} d x}}{8} - \frac{\int{4 \cos{\left(2 x \right)} \cos{\left(3 x \right)} d x}}{8} + \frac{\int{12 \cos{\left(x \right)} d x}}{8} + \frac{{\color{red}{\sin{\left(u \right)}}}}{6}$$

回忆一下 $$$u=3 x$$$:

$$- \frac{\int{12 \cos{\left(x \right)} \cos{\left(2 x \right)} d x}}{8} - \frac{\int{4 \cos{\left(2 x \right)} \cos{\left(3 x \right)} d x}}{8} + \frac{\int{12 \cos{\left(x \right)} d x}}{8} + \frac{\sin{\left({\color{red}{u}} \right)}}{6} = - \frac{\int{12 \cos{\left(x \right)} \cos{\left(2 x \right)} d x}}{8} - \frac{\int{4 \cos{\left(2 x \right)} \cos{\left(3 x \right)} d x}}{8} + \frac{\int{12 \cos{\left(x \right)} d x}}{8} + \frac{\sin{\left({\color{red}{\left(3 x\right)}} \right)}}{6}$$

对 $$$c=12$$$ 和 $$$f{\left(x \right)} = \cos{\left(x \right)}$$$ 应用常数倍法则 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$：

$$\frac{\sin{\left(3 x \right)}}{6} - \frac{\int{12 \cos{\left(x \right)} \cos{\left(2 x \right)} d x}}{8} - \frac{\int{4 \cos{\left(2 x \right)} \cos{\left(3 x \right)} d x}}{8} + \frac{{\color{red}{\int{12 \cos{\left(x \right)} d x}}}}{8} = \frac{\sin{\left(3 x \right)}}{6} - \frac{\int{12 \cos{\left(x \right)} \cos{\left(2 x \right)} d x}}{8} - \frac{\int{4 \cos{\left(2 x \right)} \cos{\left(3 x \right)} d x}}{8} + \frac{{\color{red}{\left(12 \int{\cos{\left(x \right)} d x}\right)}}}{8}$$

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

$$\frac{\sin{\left(3 x \right)}}{6} - \frac{\int{12 \cos{\left(x \right)} \cos{\left(2 x \right)} d x}}{8} - \frac{\int{4 \cos{\left(2 x \right)} \cos{\left(3 x \right)} d x}}{8} + \frac{3 {\color{red}{\int{\cos{\left(x \right)} d x}}}}{2} = \frac{\sin{\left(3 x \right)}}{6} - \frac{\int{12 \cos{\left(x \right)} \cos{\left(2 x \right)} d x}}{8} - \frac{\int{4 \cos{\left(2 x \right)} \cos{\left(3 x \right)} d x}}{8} + \frac{3 {\color{red}{\sin{\left(x \right)}}}}{2}$$

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 $$$\beta=2 x$$$，将 $$$\cos\left(x \right)\cos\left(2 x \right)$$$ 重写:

$$\frac{3 \sin{\left(x \right)}}{2} + \frac{\sin{\left(3 x \right)}}{6} - \frac{\int{4 \cos{\left(2 x \right)} \cos{\left(3 x \right)} d x}}{8} - \frac{{\color{red}{\int{12 \cos{\left(x \right)} \cos{\left(2 x \right)} d x}}}}{8} = \frac{3 \sin{\left(x \right)}}{2} + \frac{\sin{\left(3 x \right)}}{6} - \frac{\int{4 \cos{\left(2 x \right)} \cos{\left(3 x \right)} d x}}{8} - \frac{{\color{red}{\int{\left(6 \cos{\left(x \right)} + 6 \cos{\left(3 x \right)}\right)d x}}}}{8}$$

对 $$$c=\frac{1}{2}$$$ 和 $$$f{\left(x \right)} = 12 \cos{\left(x \right)} + 12 \cos{\left(3 x \right)}$$$ 应用常数倍法则 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$：

$$\frac{3 \sin{\left(x \right)}}{2} + \frac{\sin{\left(3 x \right)}}{6} - \frac{\int{4 \cos{\left(2 x \right)} \cos{\left(3 x \right)} d x}}{8} - \frac{{\color{red}{\int{\left(6 \cos{\left(x \right)} + 6 \cos{\left(3 x \right)}\right)d x}}}}{8} = \frac{3 \sin{\left(x \right)}}{2} + \frac{\sin{\left(3 x \right)}}{6} - \frac{\int{4 \cos{\left(2 x \right)} \cos{\left(3 x \right)} d x}}{8} - \frac{{\color{red}{\left(\frac{\int{\left(12 \cos{\left(x \right)} + 12 \cos{\left(3 x \right)}\right)d x}}{2}\right)}}}{8}$$

逐项积分：

$$\frac{3 \sin{\left(x \right)}}{2} + \frac{\sin{\left(3 x \right)}}{6} - \frac{\int{4 \cos{\left(2 x \right)} \cos{\left(3 x \right)} d x}}{8} - \frac{{\color{red}{\int{\left(12 \cos{\left(x \right)} + 12 \cos{\left(3 x \right)}\right)d x}}}}{16} = \frac{3 \sin{\left(x \right)}}{2} + \frac{\sin{\left(3 x \right)}}{6} - \frac{\int{4 \cos{\left(2 x \right)} \cos{\left(3 x \right)} d x}}{8} - \frac{{\color{red}{\left(\int{12 \cos{\left(x \right)} d x} + \int{12 \cos{\left(3 x \right)} d x}\right)}}}{16}$$

对 $$$c=12$$$ 和 $$$f{\left(x \right)} = \cos{\left(x \right)}$$$ 应用常数倍法则 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$：

$$\frac{3 \sin{\left(x \right)}}{2} + \frac{\sin{\left(3 x \right)}}{6} - \frac{\int{4 \cos{\left(2 x \right)} \cos{\left(3 x \right)} d x}}{8} - \frac{\int{12 \cos{\left(3 x \right)} d x}}{16} - \frac{{\color{red}{\int{12 \cos{\left(x \right)} d x}}}}{16} = \frac{3 \sin{\left(x \right)}}{2} + \frac{\sin{\left(3 x \right)}}{6} - \frac{\int{4 \cos{\left(2 x \right)} \cos{\left(3 x \right)} d x}}{8} - \frac{\int{12 \cos{\left(3 x \right)} d x}}{16} - \frac{{\color{red}{\left(12 \int{\cos{\left(x \right)} d x}\right)}}}{16}$$

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

$$\frac{3 \sin{\left(x \right)}}{2} + \frac{\sin{\left(3 x \right)}}{6} - \frac{\int{4 \cos{\left(2 x \right)} \cos{\left(3 x \right)} d x}}{8} - \frac{\int{12 \cos{\left(3 x \right)} d x}}{16} - \frac{3 {\color{red}{\int{\cos{\left(x \right)} d x}}}}{4} = \frac{3 \sin{\left(x \right)}}{2} + \frac{\sin{\left(3 x \right)}}{6} - \frac{\int{4 \cos{\left(2 x \right)} \cos{\left(3 x \right)} d x}}{8} - \frac{\int{12 \cos{\left(3 x \right)} d x}}{16} - \frac{3 {\color{red}{\sin{\left(x \right)}}}}{4}$$

对 $$$c=12$$$ 和 $$$f{\left(x \right)} = \cos{\left(3 x \right)}$$$ 应用常数倍法则 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$：

$$\frac{3 \sin{\left(x \right)}}{4} + \frac{\sin{\left(3 x \right)}}{6} - \frac{\int{4 \cos{\left(2 x \right)} \cos{\left(3 x \right)} d x}}{8} - \frac{{\color{red}{\int{12 \cos{\left(3 x \right)} d x}}}}{16} = \frac{3 \sin{\left(x \right)}}{4} + \frac{\sin{\left(3 x \right)}}{6} - \frac{\int{4 \cos{\left(2 x \right)} \cos{\left(3 x \right)} d x}}{8} - \frac{{\color{red}{\left(12 \int{\cos{\left(3 x \right)} d x}\right)}}}{16}$$

积分 $$$\int{\cos{\left(3 x \right)} d x}$$$ 已经计算过：

$$\int{\cos{\left(3 x \right)} d x} = \frac{\sin{\left(3 x \right)}}{3}$$

因此，

$$\frac{3 \sin{\left(x \right)}}{4} + \frac{\sin{\left(3 x \right)}}{6} - \frac{\int{4 \cos{\left(2 x \right)} \cos{\left(3 x \right)} d x}}{8} - \frac{3 {\color{red}{\int{\cos{\left(3 x \right)} d x}}}}{4} = \frac{3 \sin{\left(x \right)}}{4} + \frac{\sin{\left(3 x \right)}}{6} - \frac{\int{4 \cos{\left(2 x \right)} \cos{\left(3 x \right)} d x}}{8} - \frac{3 {\color{red}{\left(\frac{\sin{\left(3 x \right)}}{3}\right)}}}{4}$$

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 $$$\alpha=2 x$$$ 和 $$$\beta=3 x$$$，将 $$$\cos\left(2 x \right)\cos\left(3 x \right)$$$ 重写:

$$\frac{3 \sin{\left(x \right)}}{4} - \frac{\sin{\left(3 x \right)}}{12} - \frac{{\color{red}{\int{4 \cos{\left(2 x \right)} \cos{\left(3 x \right)} d x}}}}{8} = \frac{3 \sin{\left(x \right)}}{4} - \frac{\sin{\left(3 x \right)}}{12} - \frac{{\color{red}{\int{\left(2 \cos{\left(x \right)} + 2 \cos{\left(5 x \right)}\right)d x}}}}{8}$$

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

$$\frac{3 \sin{\left(x \right)}}{4} - \frac{\sin{\left(3 x \right)}}{12} - \frac{{\color{red}{\int{\left(2 \cos{\left(x \right)} + 2 \cos{\left(5 x \right)}\right)d x}}}}{8} = \frac{3 \sin{\left(x \right)}}{4} - \frac{\sin{\left(3 x \right)}}{12} - \frac{{\color{red}{\left(\frac{\int{\left(4 \cos{\left(x \right)} + 4 \cos{\left(5 x \right)}\right)d x}}{2}\right)}}}{8}$$

逐项积分：

$$\frac{3 \sin{\left(x \right)}}{4} - \frac{\sin{\left(3 x \right)}}{12} - \frac{{\color{red}{\int{\left(4 \cos{\left(x \right)} + 4 \cos{\left(5 x \right)}\right)d x}}}}{16} = \frac{3 \sin{\left(x \right)}}{4} - \frac{\sin{\left(3 x \right)}}{12} - \frac{{\color{red}{\left(\int{4 \cos{\left(x \right)} d x} + \int{4 \cos{\left(5 x \right)} d x}\right)}}}{16}$$

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

$$\frac{3 \sin{\left(x \right)}}{4} - \frac{\sin{\left(3 x \right)}}{12} - \frac{\int{4 \cos{\left(5 x \right)} d x}}{16} - \frac{{\color{red}{\int{4 \cos{\left(x \right)} d x}}}}{16} = \frac{3 \sin{\left(x \right)}}{4} - \frac{\sin{\left(3 x \right)}}{12} - \frac{\int{4 \cos{\left(5 x \right)} d x}}{16} - \frac{{\color{red}{\left(4 \int{\cos{\left(x \right)} d x}\right)}}}{16}$$

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

$$\frac{3 \sin{\left(x \right)}}{4} - \frac{\sin{\left(3 x \right)}}{12} - \frac{\int{4 \cos{\left(5 x \right)} d x}}{16} - \frac{{\color{red}{\int{\cos{\left(x \right)} d x}}}}{4} = \frac{3 \sin{\left(x \right)}}{4} - \frac{\sin{\left(3 x \right)}}{12} - \frac{\int{4 \cos{\left(5 x \right)} d x}}{16} - \frac{{\color{red}{\sin{\left(x \right)}}}}{4}$$

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

$$\frac{\sin{\left(x \right)}}{2} - \frac{\sin{\left(3 x \right)}}{12} - \frac{{\color{red}{\int{4 \cos{\left(5 x \right)} d x}}}}{16} = \frac{\sin{\left(x \right)}}{2} - \frac{\sin{\left(3 x \right)}}{12} - \frac{{\color{red}{\left(4 \int{\cos{\left(5 x \right)} d x}\right)}}}{16}$$

设$$$v=5 x$$$。

则$$$dv=\left(5 x\right)^{\prime }dx = 5 dx$$$ (步骤见»)，并有$$$dx = \frac{dv}{5}$$$。

因此，

$$\frac{\sin{\left(x \right)}}{2} - \frac{\sin{\left(3 x \right)}}{12} - \frac{{\color{red}{\int{\cos{\left(5 x \right)} d x}}}}{4} = \frac{\sin{\left(x \right)}}{2} - \frac{\sin{\left(3 x \right)}}{12} - \frac{{\color{red}{\int{\frac{\cos{\left(v \right)}}{5} d v}}}}{4}$$

对 $$$c=\frac{1}{5}$$$ 和 $$$f{\left(v \right)} = \cos{\left(v \right)}$$$ 应用常数倍法则 $$$\int c f{\left(v \right)}\, dv = c \int f{\left(v \right)}\, dv$$$：

$$\frac{\sin{\left(x \right)}}{2} - \frac{\sin{\left(3 x \right)}}{12} - \frac{{\color{red}{\int{\frac{\cos{\left(v \right)}}{5} d v}}}}{4} = \frac{\sin{\left(x \right)}}{2} - \frac{\sin{\left(3 x \right)}}{12} - \frac{{\color{red}{\left(\frac{\int{\cos{\left(v \right)} d v}}{5}\right)}}}{4}$$

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

$$\frac{\sin{\left(x \right)}}{2} - \frac{\sin{\left(3 x \right)}}{12} - \frac{{\color{red}{\int{\cos{\left(v \right)} d v}}}}{20} = \frac{\sin{\left(x \right)}}{2} - \frac{\sin{\left(3 x \right)}}{12} - \frac{{\color{red}{\sin{\left(v \right)}}}}{20}$$

回忆一下 $$$v=5 x$$$:

$$\frac{\sin{\left(x \right)}}{2} - \frac{\sin{\left(3 x \right)}}{12} - \frac{\sin{\left({\color{red}{v}} \right)}}{20} = \frac{\sin{\left(x \right)}}{2} - \frac{\sin{\left(3 x \right)}}{12} - \frac{\sin{\left({\color{red}{\left(5 x\right)}} \right)}}{20}$$

因此，

$$\int{4 \sin^{2}{\left(x \right)} \cos^{3}{\left(x \right)} d x} = \frac{\sin{\left(x \right)}}{2} - \frac{\sin{\left(3 x \right)}}{12} - \frac{\sin{\left(5 x \right)}}{20}$$

加上积分常数：

$$\int{4 \sin^{2}{\left(x \right)} \cos^{3}{\left(x \right)} d x} = \frac{\sin{\left(x \right)}}{2} - \frac{\sin{\left(3 x \right)}}{12} - \frac{\sin{\left(5 x \right)}}{20}+C$$

$$$\int 4 \sin^{2}{\left(x \right)} \cos^{3}{\left(x \right)}\, dx = \left(\frac{\sin{\left(x \right)}}{2} - \frac{\sin{\left(3 x \right)}}{12} - \frac{\sin{\left(5 x \right)}}{20}\right) + C$$$A