$$$4 \cos^{4}{\left(\frac{\theta}{2} \right)}$$$ 的积分

求$$$\int 4 \cos^{4}{\left(\frac{\theta}{2} \right)}\, d\theta$$$。

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

$${\color{red}{\int{4 \cos^{4}{\left(\frac{\theta}{2} \right)} d \theta}}} = {\color{red}{\left(4 \int{\cos^{4}{\left(\frac{\theta}{2} \right)} d \theta}\right)}}$$

设$$$u=\frac{\theta}{2}$$$。

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

所以，

$$4 {\color{red}{\int{\cos^{4}{\left(\frac{\theta}{2} \right)} d \theta}}} = 4 {\color{red}{\int{2 \cos^{4}{\left(u \right)} d u}}}$$

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

$$4 {\color{red}{\int{2 \cos^{4}{\left(u \right)} d u}}} = 4 {\color{red}{\left(2 \int{\cos^{4}{\left(u \right)} d u}\right)}}$$

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 $$$\alpha= u $$$:

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

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 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$：

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

逐项积分：

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

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

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

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

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

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

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

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

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

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

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

回忆一下 $$$v=2 u$$$:

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

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，

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

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

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

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

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

回忆一下 $$$v=4 u$$$:

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

回忆一下 $$$u=\frac{\theta}{2}$$$:

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

因此，

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

化简：

$$\int{4 \cos^{4}{\left(\frac{\theta}{2} \right)} d \theta} = \frac{6 \theta + 8 \sin{\left(\theta \right)} + \sin{\left(2 \theta \right)}}{4}$$

加上积分常数：

$$\int{4 \cos^{4}{\left(\frac{\theta}{2} \right)} d \theta} = \frac{6 \theta + 8 \sin{\left(\theta \right)} + \sin{\left(2 \theta \right)}}{4}+C$$

$$$\int 4 \cos^{4}{\left(\frac{\theta}{2} \right)}\, d\theta = \frac{6 \theta + 8 \sin{\left(\theta \right)} + \sin{\left(2 \theta \right)}}{4} + C$$$A