$$$\cos^{2}{\left(c \right)}$$$ 的積分

求$$$\int \cos^{2}{\left(c \right)}\, dc$$$。

套用降冪公式 $$$\cos^{2}{\left(\alpha \right)} = \frac{\cos{\left(2 \alpha \right)}}{2} + \frac{1}{2}$$$，令 $$$\alpha=c$$$:

$${\color{red}{\int{\cos^{2}{\left(c \right)} d c}}} = {\color{red}{\int{\left(\frac{\cos{\left(2 c \right)}}{2} + \frac{1}{2}\right)d c}}}$$

套用常數倍法則 $$$\int c f{\left(c \right)}\, dc = c \int f{\left(c \right)}\, dc$$$，使用 $$$c=\frac{1}{2}$$$ 與 $$$f{\left(c \right)} = \cos{\left(2 c \right)} + 1$$$：

$${\color{red}{\int{\left(\frac{\cos{\left(2 c \right)}}{2} + \frac{1}{2}\right)d c}}} = {\color{red}{\left(\frac{\int{\left(\cos{\left(2 c \right)} + 1\right)d c}}{2}\right)}}$$

逐項積分：

$$\frac{{\color{red}{\int{\left(\cos{\left(2 c \right)} + 1\right)d c}}}}{2} = \frac{{\color{red}{\left(\int{1 d c} + \int{\cos{\left(2 c \right)} d c}\right)}}}{2}$$

配合 $$$c=1$$$，應用常數法則 $$$\int c\, dc = c c$$$：

$$\frac{\int{\cos{\left(2 c \right)} d c}}{2} + \frac{{\color{red}{\int{1 d c}}}}{2} = \frac{\int{\cos{\left(2 c \right)} d c}}{2} + \frac{{\color{red}{c}}}{2}$$

令 $$$u=2 c$$$。

則 $$$du=\left(2 c\right)^{\prime }dc = 2 dc$$$ (步驟見»)，並可得 $$$dc = \frac{du}{2}$$$。

因此，

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

套用常數倍法則 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$，使用 $$$c=\frac{1}{2}$$$ 與 $$$f{\left(u \right)} = \cos{\left(u \right)}$$$：

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

餘弦函數的積分為 $$$\int{\cos{\left(u \right)} d u} = \sin{\left(u \right)}$$$：

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

回顧一下 $$$u=2 c$$$：

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

因此，

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

加上積分常數：

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

$$$\int \cos^{2}{\left(c \right)}\, dc = \left(\frac{c}{2} + \frac{\sin{\left(2 c \right)}}{4}\right) + C$$$A