$$$36 \cos^{2}{\left(\theta \right)}$$$ 的積分

求$$$\int 36 \cos^{2}{\left(\theta \right)}\, d\theta$$$。

套用常數倍法則 $$$\int c f{\left(\theta \right)}\, d\theta = c \int f{\left(\theta \right)}\, d\theta$$$，使用 $$$c=36$$$ 與 $$$f{\left(\theta \right)} = \cos^{2}{\left(\theta \right)}$$$：

$${\color{red}{\int{36 \cos^{2}{\left(\theta \right)} d \theta}}} = {\color{red}{\left(36 \int{\cos^{2}{\left(\theta \right)} d \theta}\right)}}$$

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

$$36 {\color{red}{\int{\cos^{2}{\left(\theta \right)} d \theta}}} = 36 {\color{red}{\int{\left(\frac{\cos{\left(2 \theta \right)}}{2} + \frac{1}{2}\right)d \theta}}}$$

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

$$36 {\color{red}{\int{\left(\frac{\cos{\left(2 \theta \right)}}{2} + \frac{1}{2}\right)d \theta}}} = 36 {\color{red}{\left(\frac{\int{\left(\cos{\left(2 \theta \right)} + 1\right)d \theta}}{2}\right)}}$$

逐項積分：

$$18 {\color{red}{\int{\left(\cos{\left(2 \theta \right)} + 1\right)d \theta}}} = 18 {\color{red}{\left(\int{1 d \theta} + \int{\cos{\left(2 \theta \right)} d \theta}\right)}}$$

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

$$18 \int{\cos{\left(2 \theta \right)} d \theta} + 18 {\color{red}{\int{1 d \theta}}} = 18 \int{\cos{\left(2 \theta \right)} d \theta} + 18 {\color{red}{\theta}}$$

令 $$$u=2 \theta$$$。

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

所以，

$$18 \theta + 18 {\color{red}{\int{\cos{\left(2 \theta \right)} d \theta}}} = 18 \theta + 18 {\color{red}{\int{\frac{\cos{\left(u \right)}}{2} d u}}}$$

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 $$$f{\left(u \right)} = \cos{\left(u \right)}$$$：

$$18 \theta + 18 {\color{red}{\int{\frac{\cos{\left(u \right)}}{2} d u}}} = 18 \theta + 18 {\color{red}{\left(\frac{\int{\cos{\left(u \right)} d u}}{2}\right)}}$$

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

$$18 \theta + 9 {\color{red}{\int{\cos{\left(u \right)} d u}}} = 18 \theta + 9 {\color{red}{\sin{\left(u \right)}}}$$

回顧一下 $$$u=2 \theta$$$：

$$18 \theta + 9 \sin{\left({\color{red}{u}} \right)} = 18 \theta + 9 \sin{\left({\color{red}{\left(2 \theta\right)}} \right)}$$

因此，

$$\int{36 \cos^{2}{\left(\theta \right)} d \theta} = 18 \theta + 9 \sin{\left(2 \theta \right)}$$

加上積分常數：

$$\int{36 \cos^{2}{\left(\theta \right)} d \theta} = 18 \theta + 9 \sin{\left(2 \theta \right)}+C$$

$$$\int 36 \cos^{2}{\left(\theta \right)}\, d\theta = \left(18 \theta + 9 \sin{\left(2 \theta \right)}\right) + C$$$A