$$$\sin^{4}{\left(3 x \right)}$$$ 的积分

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

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，

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

对 $$$c=\frac{1}{3}$$$ 和 $$$f{\left(u \right)} = \sin^{4}{\left(u \right)}$$$ 应用常数倍法则 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$：

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

应用降幂公式 $$$\sin^{4}{\left(\alpha \right)} = - \frac{\cos{\left(2 \alpha \right)}}{2} + \frac{\cos{\left(4 \alpha \right)}}{8} + \frac{3}{8}$$$，并令 $$$\alpha= u $$$:

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

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

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

逐项积分：

$$\frac{{\color{red}{\int{\left(- 4 \cos{\left(2 u \right)} + \cos{\left(4 u \right)} + 3\right)d u}}}}{24} = \frac{{\color{red}{\left(\int{3 d u} - \int{4 \cos{\left(2 u \right)} d u} + \int{\cos{\left(4 u \right)} d u}\right)}}}{24}$$

应用常数法则 $$$\int c\, du = c u$$$，使用 $$$c=3$$$：

$$- \frac{\int{4 \cos{\left(2 u \right)} d u}}{24} + \frac{\int{\cos{\left(4 u \right)} d u}}{24} + \frac{{\color{red}{\int{3 d u}}}}{24} = - \frac{\int{4 \cos{\left(2 u \right)} d u}}{24} + \frac{\int{\cos{\left(4 u \right)} d u}}{24} + \frac{{\color{red}{\left(3 u\right)}}}{24}$$

对 $$$c=4$$$ 和 $$$f{\left(u \right)} = \cos{\left(2 u \right)}$$$ 应用常数倍法则 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$：

$$\frac{u}{8} + \frac{\int{\cos{\left(4 u \right)} d u}}{24} - \frac{{\color{red}{\int{4 \cos{\left(2 u \right)} d u}}}}{24} = \frac{u}{8} + \frac{\int{\cos{\left(4 u \right)} d u}}{24} - \frac{{\color{red}{\left(4 \int{\cos{\left(2 u \right)} d u}\right)}}}{24}$$

设$$$v=2 u$$$。

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

积分变为

$$\frac{u}{8} + \frac{\int{\cos{\left(4 u \right)} d u}}{24} - \frac{{\color{red}{\int{\cos{\left(2 u \right)} d u}}}}{6} = \frac{u}{8} + \frac{\int{\cos{\left(4 u \right)} d u}}{24} - \frac{{\color{red}{\int{\frac{\cos{\left(v \right)}}{2} d v}}}}{6}$$

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

$$\frac{u}{8} + \frac{\int{\cos{\left(4 u \right)} d u}}{24} - \frac{{\color{red}{\int{\frac{\cos{\left(v \right)}}{2} d v}}}}{6} = \frac{u}{8} + \frac{\int{\cos{\left(4 u \right)} d u}}{24} - \frac{{\color{red}{\left(\frac{\int{\cos{\left(v \right)} d v}}{2}\right)}}}{6}$$

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

$$\frac{u}{8} + \frac{\int{\cos{\left(4 u \right)} d u}}{24} - \frac{{\color{red}{\int{\cos{\left(v \right)} d v}}}}{12} = \frac{u}{8} + \frac{\int{\cos{\left(4 u \right)} d u}}{24} - \frac{{\color{red}{\sin{\left(v \right)}}}}{12}$$

回忆一下 $$$v=2 u$$$:

$$\frac{u}{8} + \frac{\int{\cos{\left(4 u \right)} d u}}{24} - \frac{\sin{\left({\color{red}{v}} \right)}}{12} = \frac{u}{8} + \frac{\int{\cos{\left(4 u \right)} d u}}{24} - \frac{\sin{\left({\color{red}{\left(2 u\right)}} \right)}}{12}$$

设$$$v=4 u$$$。

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

因此，

$$\frac{u}{8} - \frac{\sin{\left(2 u \right)}}{12} + \frac{{\color{red}{\int{\cos{\left(4 u \right)} d u}}}}{24} = \frac{u}{8} - \frac{\sin{\left(2 u \right)}}{12} + \frac{{\color{red}{\int{\frac{\cos{\left(v \right)}}{4} d v}}}}{24}$$

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

$$\frac{u}{8} - \frac{\sin{\left(2 u \right)}}{12} + \frac{{\color{red}{\int{\frac{\cos{\left(v \right)}}{4} d v}}}}{24} = \frac{u}{8} - \frac{\sin{\left(2 u \right)}}{12} + \frac{{\color{red}{\left(\frac{\int{\cos{\left(v \right)} d v}}{4}\right)}}}{24}$$

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

$$\frac{u}{8} - \frac{\sin{\left(2 u \right)}}{12} + \frac{{\color{red}{\int{\cos{\left(v \right)} d v}}}}{96} = \frac{u}{8} - \frac{\sin{\left(2 u \right)}}{12} + \frac{{\color{red}{\sin{\left(v \right)}}}}{96}$$

回忆一下 $$$v=4 u$$$:

$$\frac{u}{8} - \frac{\sin{\left(2 u \right)}}{12} + \frac{\sin{\left({\color{red}{v}} \right)}}{96} = \frac{u}{8} - \frac{\sin{\left(2 u \right)}}{12} + \frac{\sin{\left({\color{red}{\left(4 u\right)}} \right)}}{96}$$

回忆一下 $$$u=3 x$$$:

$$- \frac{\sin{\left(2 {\color{red}{u}} \right)}}{12} + \frac{\sin{\left(4 {\color{red}{u}} \right)}}{96} + \frac{{\color{red}{u}}}{8} = - \frac{\sin{\left(2 {\color{red}{\left(3 x\right)}} \right)}}{12} + \frac{\sin{\left(4 {\color{red}{\left(3 x\right)}} \right)}}{96} + \frac{{\color{red}{\left(3 x\right)}}}{8}$$

因此，

$$\int{\sin^{4}{\left(3 x \right)} d x} = \frac{3 x}{8} - \frac{\sin{\left(6 x \right)}}{12} + \frac{\sin{\left(12 x \right)}}{96}$$

化简：

$$\int{\sin^{4}{\left(3 x \right)} d x} = \frac{36 x - 8 \sin{\left(6 x \right)} + \sin{\left(12 x \right)}}{96}$$

加上积分常数：

$$\int{\sin^{4}{\left(3 x \right)} d x} = \frac{36 x - 8 \sin{\left(6 x \right)} + \sin{\left(12 x \right)}}{96}+C$$

$$$\int \sin^{4}{\left(3 x \right)}\, dx = \frac{36 x - 8 \sin{\left(6 x \right)} + \sin{\left(12 x \right)}}{96} + C$$$A