$$$\frac{\left(1 - \frac{\sin{\left(t \right)}}{2}\right)^{2} \cos{\left(t \right)}}{2}$$$ 的積分

求$$$\int \frac{\left(1 - \frac{\sin{\left(t \right)}}{2}\right)^{2} \cos{\left(t \right)}}{2}\, dt$$$。

套用常數倍法則 $$$\int c f{\left(t \right)}\, dt = c \int f{\left(t \right)}\, dt$$$，使用 $$$c=\frac{1}{2}$$$ 與 $$$f{\left(t \right)} = \left(1 - \frac{\sin{\left(t \right)}}{2}\right)^{2} \cos{\left(t \right)}$$$：

$${\color{red}{\int{\frac{\left(1 - \frac{\sin{\left(t \right)}}{2}\right)^{2} \cos{\left(t \right)}}{2} d t}}} = {\color{red}{\left(\frac{\int{\left(1 - \frac{\sin{\left(t \right)}}{2}\right)^{2} \cos{\left(t \right)} d t}}{2}\right)}}$$

令 $$$u=1 - \frac{\sin{\left(t \right)}}{2}$$$。

則 $$$du=\left(1 - \frac{\sin{\left(t \right)}}{2}\right)^{\prime }dt = - \frac{\cos{\left(t \right)}}{2} dt$$$ (步驟見»)，並可得 $$$\cos{\left(t \right)} dt = - 2 du$$$。

該積分變為

$$\frac{{\color{red}{\int{\left(1 - \frac{\sin{\left(t \right)}}{2}\right)^{2} \cos{\left(t \right)} d t}}}}{2} = \frac{{\color{red}{\int{\left(- 2 u^{2}\right)d u}}}}{2}$$

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

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

套用冪次法則 $$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$，以 $$$n=2$$$：

$$- {\color{red}{\int{u^{2} d u}}}=- {\color{red}{\frac{u^{1 + 2}}{1 + 2}}}=- {\color{red}{\left(\frac{u^{3}}{3}\right)}}$$

回顧一下 $$$u=1 - \frac{\sin{\left(t \right)}}{2}$$$：

$$- \frac{{\color{red}{u}}^{3}}{3} = - \frac{{\color{red}{\left(1 - \frac{\sin{\left(t \right)}}{2}\right)}}^{3}}{3}$$

因此，

$$\int{\frac{\left(1 - \frac{\sin{\left(t \right)}}{2}\right)^{2} \cos{\left(t \right)}}{2} d t} = - \frac{\left(1 - \frac{\sin{\left(t \right)}}{2}\right)^{3}}{3}$$

化簡：

$$\int{\frac{\left(1 - \frac{\sin{\left(t \right)}}{2}\right)^{2} \cos{\left(t \right)}}{2} d t} = \frac{\left(\sin{\left(t \right)} - 2\right)^{3}}{24}$$

加上積分常數：

$$\int{\frac{\left(1 - \frac{\sin{\left(t \right)}}{2}\right)^{2} \cos{\left(t \right)}}{2} d t} = \frac{\left(\sin{\left(t \right)} - 2\right)^{3}}{24}+C$$

$$$\int \frac{\left(1 - \frac{\sin{\left(t \right)}}{2}\right)^{2} \cos{\left(t \right)}}{2}\, dt = \frac{\left(\sin{\left(t \right)} - 2\right)^{3}}{24} + C$$$A