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

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

逐项积分：

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

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

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

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

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

逐项积分：

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

应用常数法则 $$$\int c\, dx = c x$$$，使用 $$$c=3$$$：

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

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

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

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，

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

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

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

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

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

回忆一下 $$$u=2 x$$$:

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

设$$$u=4 x$$$。

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

该积分可以改写为

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

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

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

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

$$\frac{3 x}{8} + \frac{\sin{\left(2 x \right)}}{4} + \int{\sin^{4}{\left(x \right)} d x} + \frac{{\color{red}{\int{\cos{\left(u \right)} d u}}}}{32} = \frac{3 x}{8} + \frac{\sin{\left(2 x \right)}}{4} + \int{\sin^{4}{\left(x \right)} d x} + \frac{{\color{red}{\sin{\left(u \right)}}}}{32}$$

回忆一下 $$$u=4 x$$$:

$$\frac{3 x}{8} + \frac{\sin{\left(2 x \right)}}{4} + \int{\sin^{4}{\left(x \right)} d x} + \frac{\sin{\left({\color{red}{u}} \right)}}{32} = \frac{3 x}{8} + \frac{\sin{\left(2 x \right)}}{4} + \int{\sin^{4}{\left(x \right)} d x} + \frac{\sin{\left({\color{red}{\left(4 x\right)}} \right)}}{32}$$

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 $$$\alpha=x$$$:

$$\frac{3 x}{8} + \frac{\sin{\left(2 x \right)}}{4} + \frac{\sin{\left(4 x \right)}}{32} + {\color{red}{\int{\sin^{4}{\left(x \right)} d x}}} = \frac{3 x}{8} + \frac{\sin{\left(2 x \right)}}{4} + \frac{\sin{\left(4 x \right)}}{32} + {\color{red}{\int{\left(- \frac{\cos{\left(2 x \right)}}{2} + \frac{\cos{\left(4 x \right)}}{8} + \frac{3}{8}\right)d x}}}$$

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

$$\frac{3 x}{8} + \frac{\sin{\left(2 x \right)}}{4} + \frac{\sin{\left(4 x \right)}}{32} + {\color{red}{\int{\left(- \frac{\cos{\left(2 x \right)}}{2} + \frac{\cos{\left(4 x \right)}}{8} + \frac{3}{8}\right)d x}}} = \frac{3 x}{8} + \frac{\sin{\left(2 x \right)}}{4} + \frac{\sin{\left(4 x \right)}}{32} + {\color{red}{\left(\frac{\int{\left(- 4 \cos{\left(2 x \right)} + \cos{\left(4 x \right)} + 3\right)d x}}{8}\right)}}$$

逐项积分：

$$\frac{3 x}{8} + \frac{\sin{\left(2 x \right)}}{4} + \frac{\sin{\left(4 x \right)}}{32} + \frac{{\color{red}{\int{\left(- 4 \cos{\left(2 x \right)} + \cos{\left(4 x \right)} + 3\right)d x}}}}{8} = \frac{3 x}{8} + \frac{\sin{\left(2 x \right)}}{4} + \frac{\sin{\left(4 x \right)}}{32} + \frac{{\color{red}{\left(\int{3 d x} - \int{4 \cos{\left(2 x \right)} d x} + \int{\cos{\left(4 x \right)} d x}\right)}}}{8}$$

应用常数法则 $$$\int c\, dx = c x$$$，使用 $$$c=3$$$：

$$\frac{3 x}{8} + \frac{\sin{\left(2 x \right)}}{4} + \frac{\sin{\left(4 x \right)}}{32} - \frac{\int{4 \cos{\left(2 x \right)} d x}}{8} + \frac{\int{\cos{\left(4 x \right)} d x}}{8} + \frac{{\color{red}{\int{3 d x}}}}{8} = \frac{3 x}{8} + \frac{\sin{\left(2 x \right)}}{4} + \frac{\sin{\left(4 x \right)}}{32} - \frac{\int{4 \cos{\left(2 x \right)} d x}}{8} + \frac{\int{\cos{\left(4 x \right)} d x}}{8} + \frac{{\color{red}{\left(3 x\right)}}}{8}$$

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

$$\frac{3 x}{4} + \frac{\sin{\left(2 x \right)}}{4} + \frac{\sin{\left(4 x \right)}}{32} + \frac{\int{\cos{\left(4 x \right)} d x}}{8} - \frac{{\color{red}{\int{4 \cos{\left(2 x \right)} d x}}}}{8} = \frac{3 x}{4} + \frac{\sin{\left(2 x \right)}}{4} + \frac{\sin{\left(4 x \right)}}{32} + \frac{\int{\cos{\left(4 x \right)} d x}}{8} - \frac{{\color{red}{\left(4 \int{\cos{\left(2 x \right)} d x}\right)}}}{8}$$

积分 $$$\int{\cos{\left(2 x \right)} d x}$$$ 已经计算过：

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

因此，

$$\frac{3 x}{4} + \frac{\sin{\left(2 x \right)}}{4} + \frac{\sin{\left(4 x \right)}}{32} + \frac{\int{\cos{\left(4 x \right)} d x}}{8} - \frac{{\color{red}{\int{\cos{\left(2 x \right)} d x}}}}{2} = \frac{3 x}{4} + \frac{\sin{\left(2 x \right)}}{4} + \frac{\sin{\left(4 x \right)}}{32} + \frac{\int{\cos{\left(4 x \right)} d x}}{8} - \frac{{\color{red}{\left(\frac{\sin{\left(2 x \right)}}{2}\right)}}}{2}$$

积分 $$$\int{\cos{\left(4 x \right)} d x}$$$ 已经计算过：

$$\int{\cos{\left(4 x \right)} d x} = \frac{\sin{\left(4 x \right)}}{4}$$

因此，

$$\frac{3 x}{4} + \frac{\sin{\left(4 x \right)}}{32} + \frac{{\color{red}{\int{\cos{\left(4 x \right)} d x}}}}{8} = \frac{3 x}{4} + \frac{\sin{\left(4 x \right)}}{32} + \frac{{\color{red}{\left(\frac{\sin{\left(4 x \right)}}{4}\right)}}}{8}$$

因此，

$$\int{\left(\sin^{4}{\left(x \right)} + \cos^{4}{\left(x \right)}\right)d x} = \frac{3 x}{4} + \frac{\sin{\left(4 x \right)}}{16}$$

化简：

$$\int{\left(\sin^{4}{\left(x \right)} + \cos^{4}{\left(x \right)}\right)d x} = \frac{12 x + \sin{\left(4 x \right)}}{16}$$

加上积分常数：

$$\int{\left(\sin^{4}{\left(x \right)} + \cos^{4}{\left(x \right)}\right)d x} = \frac{12 x + \sin{\left(4 x \right)}}{16}+C$$

$$$\int \left(\sin^{4}{\left(x \right)} + \cos^{4}{\left(x \right)}\right)\, dx = \frac{12 x + \sin{\left(4 x \right)}}{16} + C$$$A