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

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

逐项积分：

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

提出一个余弦，并使用公式 $$$\cos^2\left(\alpha \right)=-\sin^2\left(\alpha \right)+1$$$（令 $$$\alpha=x$$$）将其余部分用正弦表示:

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

设$$$u=\sin{\left(x \right)}$$$。

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

所以，

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

逐项积分：

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

应用常数法则 $$$\int c\, du = c u$$$，使用 $$$c=1$$$：

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

应用幂法则 $$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$，其中 $$$n=2$$$：

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

回忆一下 $$$u=\sin{\left(x \right)}$$$:

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

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

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

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

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

Expand the expression:

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

逐项积分：

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

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

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

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

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

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:

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

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

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

逐项积分：

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

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

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

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

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

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

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

设$$$u=3 x$$$。

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

因此，

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

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

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

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

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

回忆一下 $$$u=3 x$$$:

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

因此，

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

化简：

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

加上积分常数：

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

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