$$$\cos^{2}{\left(c \right)}$$$ 的积分
您的输入
求$$$\int \cos^{2}{\left(c \right)}\, dc$$$。
解答
应用降幂公式 $$$\cos^{2}{\left(\alpha \right)} = \frac{\cos{\left(2 \alpha \right)}}{2} + \frac{1}{2}$$$，并令 $$$\alpha=c$$$:
$${\color{red}{\int{\cos^{2}{\left(c \right)} d c}}} = {\color{red}{\int{\left(\frac{\cos{\left(2 c \right)}}{2} + \frac{1}{2}\right)d c}}}$$
对 $$$c=\frac{1}{2}$$$ 和 $$$f{\left(c \right)} = \cos{\left(2 c \right)} + 1$$$ 应用常数倍法则 $$$\int c f{\left(c \right)}\, dc = c \int f{\left(c \right)}\, dc$$$：
$${\color{red}{\int{\left(\frac{\cos{\left(2 c \right)}}{2} + \frac{1}{2}\right)d c}}} = {\color{red}{\left(\frac{\int{\left(\cos{\left(2 c \right)} + 1\right)d c}}{2}\right)}}$$
逐项积分：
$$\frac{{\color{red}{\int{\left(\cos{\left(2 c \right)} + 1\right)d c}}}}{2} = \frac{{\color{red}{\left(\int{1 d c} + \int{\cos{\left(2 c \right)} d c}\right)}}}{2}$$
应用常数法则 $$$\int c\, dc = c c$$$，使用 $$$c=1$$$：
$$\frac{\int{\cos{\left(2 c \right)} d c}}{2} + \frac{{\color{red}{\int{1 d c}}}}{2} = \frac{\int{\cos{\left(2 c \right)} d c}}{2} + \frac{{\color{red}{c}}}{2}$$
设$$$u=2 c$$$。
则$$$du=\left(2 c\right)^{\prime }dc = 2 dc$$$ (步骤见»)，并有$$$dc = \frac{du}{2}$$$。
积分变为
$$\frac{c}{2} + \frac{{\color{red}{\int{\cos{\left(2 c \right)} d c}}}}{2} = \frac{c}{2} + \frac{{\color{red}{\int{\frac{\cos{\left(u \right)}}{2} d u}}}}{2}$$
对 $$$c=\frac{1}{2}$$$ 和 $$$f{\left(u \right)} = \cos{\left(u \right)}$$$ 应用常数倍法则 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$：
$$\frac{c}{2} + \frac{{\color{red}{\int{\frac{\cos{\left(u \right)}}{2} d u}}}}{2} = \frac{c}{2} + \frac{{\color{red}{\left(\frac{\int{\cos{\left(u \right)} d u}}{2}\right)}}}{2}$$
余弦函数的积分为 $$$\int{\cos{\left(u \right)} d u} = \sin{\left(u \right)}$$$：
$$\frac{c}{2} + \frac{{\color{red}{\int{\cos{\left(u \right)} d u}}}}{4} = \frac{c}{2} + \frac{{\color{red}{\sin{\left(u \right)}}}}{4}$$
回忆一下 $$$u=2 c$$$:
$$\frac{c}{2} + \frac{\sin{\left({\color{red}{u}} \right)}}{4} = \frac{c}{2} + \frac{\sin{\left({\color{red}{\left(2 c\right)}} \right)}}{4}$$
因此，
$$\int{\cos^{2}{\left(c \right)} d c} = \frac{c}{2} + \frac{\sin{\left(2 c \right)}}{4}$$
加上积分常数：
$$\int{\cos^{2}{\left(c \right)} d c} = \frac{c}{2} + \frac{\sin{\left(2 c \right)}}{4}+C$$
答案
$$$\int \cos^{2}{\left(c \right)}\, dc = \left(\frac{c}{2} + \frac{\sin{\left(2 c \right)}}{4}\right) + C$$$A