$$$\sin^{2}{\left(t \right)} \cos^{4}{\left(t \right)}$$$ 的積分

求$$$\int \sin^{2}{\left(t \right)} \cos^{4}{\left(t \right)}\, dt$$$。

使用降冪公式 $$$\sin^2\left( \alpha \right)=\frac{1}{2}-\frac{1}{2}\cos\left(2 \alpha \right)-$$$（參數為 $$$\alpha=t$$$）與 $$$\cos^2\left( \beta \right)=\frac{1}{2}+\frac{1}{2}\cos\left(2 \beta \right)+$$$（參數為 $$$\beta=t$$$），將被積函數改寫。:

$${\color{red}{\int{\sin^{2}{\left(t \right)} \cos^{4}{\left(t \right)} d t}}} = {\color{red}{\int{\left(\frac{1}{2} - \frac{\cos{\left(2 t \right)}}{2}\right) \left(\frac{\cos{\left(2 t \right)}}{2} + \frac{1}{2}\right)^{2} d t}}}$$

展開該表達式:

$${\color{red}{\int{\left(\frac{1}{2} - \frac{\cos{\left(2 t \right)}}{2}\right) \left(\frac{\cos{\left(2 t \right)}}{2} + \frac{1}{2}\right)^{2} d t}}} = {\color{red}{\int{\left(- \frac{\cos^{3}{\left(2 t \right)}}{8} - \frac{\cos^{2}{\left(2 t \right)}}{8} + \frac{\cos{\left(2 t \right)}}{8} + \frac{1}{8}\right)d t}}}$$

逐項積分：

$${\color{red}{\int{\left(- \frac{\cos^{3}{\left(2 t \right)}}{8} - \frac{\cos^{2}{\left(2 t \right)}}{8} + \frac{\cos{\left(2 t \right)}}{8} + \frac{1}{8}\right)d t}}} = {\color{red}{\left(\int{\frac{1}{8} d t} + \int{\frac{\cos{\left(2 t \right)}}{8} d t} - \int{\frac{\cos^{2}{\left(2 t \right)}}{8} d t} - \int{\frac{\cos^{3}{\left(2 t \right)}}{8} d t}\right)}}$$

配合 $$$c=\frac{1}{8}$$$，應用常數法則 $$$\int c\, dt = c t$$$：

$$\int{\frac{\cos{\left(2 t \right)}}{8} d t} - \int{\frac{\cos^{2}{\left(2 t \right)}}{8} d t} - \int{\frac{\cos^{3}{\left(2 t \right)}}{8} d t} + {\color{red}{\int{\frac{1}{8} d t}}} = \int{\frac{\cos{\left(2 t \right)}}{8} d t} - \int{\frac{\cos^{2}{\left(2 t \right)}}{8} d t} - \int{\frac{\cos^{3}{\left(2 t \right)}}{8} d t} + {\color{red}{\left(\frac{t}{8}\right)}}$$

套用常數倍法則 $$$\int c f{\left(t \right)}\, dt = c \int f{\left(t \right)}\, dt$$$，使用 $$$c=\frac{1}{8}$$$ 與 $$$f{\left(t \right)} = \cos^{2}{\left(2 t \right)}$$$：

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

令 $$$u=2 t$$$。

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

因此，

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

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 $$$f{\left(u \right)} = \cos^{2}{\left(u \right)}$$$：

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

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

$$\frac{t}{8} + \int{\frac{\cos{\left(2 t \right)}}{8} d t} - \int{\frac{\cos^{3}{\left(2 t \right)}}{8} d t} - \frac{{\color{red}{\int{\cos^{2}{\left(u \right)} d u}}}}{16} = \frac{t}{8} + \int{\frac{\cos{\left(2 t \right)}}{8} d t} - \int{\frac{\cos^{3}{\left(2 t \right)}}{8} d t} - \frac{{\color{red}{\int{\left(\frac{\cos{\left(2 u \right)}}{2} + \frac{1}{2}\right)d u}}}}{16}$$

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 $$$f{\left(u \right)} = \cos{\left(2 u \right)} + 1$$$：

$$\frac{t}{8} + \int{\frac{\cos{\left(2 t \right)}}{8} d t} - \int{\frac{\cos^{3}{\left(2 t \right)}}{8} d t} - \frac{{\color{red}{\int{\left(\frac{\cos{\left(2 u \right)}}{2} + \frac{1}{2}\right)d u}}}}{16} = \frac{t}{8} + \int{\frac{\cos{\left(2 t \right)}}{8} d t} - \int{\frac{\cos^{3}{\left(2 t \right)}}{8} d t} - \frac{{\color{red}{\left(\frac{\int{\left(\cos{\left(2 u \right)} + 1\right)d u}}{2}\right)}}}{16}$$

逐項積分：

$$\frac{t}{8} + \int{\frac{\cos{\left(2 t \right)}}{8} d t} - \int{\frac{\cos^{3}{\left(2 t \right)}}{8} d t} - \frac{{\color{red}{\int{\left(\cos{\left(2 u \right)} + 1\right)d u}}}}{32} = \frac{t}{8} + \int{\frac{\cos{\left(2 t \right)}}{8} d t} - \int{\frac{\cos^{3}{\left(2 t \right)}}{8} d t} - \frac{{\color{red}{\left(\int{1 d u} + \int{\cos{\left(2 u \right)} d u}\right)}}}{32}$$

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

$$\frac{t}{8} + \int{\frac{\cos{\left(2 t \right)}}{8} d t} - \int{\frac{\cos^{3}{\left(2 t \right)}}{8} d t} - \frac{\int{\cos{\left(2 u \right)} d u}}{32} - \frac{{\color{red}{\int{1 d u}}}}{32} = \frac{t}{8} + \int{\frac{\cos{\left(2 t \right)}}{8} d t} - \int{\frac{\cos^{3}{\left(2 t \right)}}{8} d t} - \frac{\int{\cos{\left(2 u \right)} d u}}{32} - \frac{{\color{red}{u}}}{32}$$

令 $$$v=2 u$$$。

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

該積分可改寫為

$$\frac{t}{8} - \frac{u}{32} + \int{\frac{\cos{\left(2 t \right)}}{8} d t} - \int{\frac{\cos^{3}{\left(2 t \right)}}{8} d t} - \frac{{\color{red}{\int{\cos{\left(2 u \right)} d u}}}}{32} = \frac{t}{8} - \frac{u}{32} + \int{\frac{\cos{\left(2 t \right)}}{8} d t} - \int{\frac{\cos^{3}{\left(2 t \right)}}{8} d t} - \frac{{\color{red}{\int{\frac{\cos{\left(v \right)}}{2} d v}}}}{32}$$

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

$$\frac{t}{8} - \frac{u}{32} + \int{\frac{\cos{\left(2 t \right)}}{8} d t} - \int{\frac{\cos^{3}{\left(2 t \right)}}{8} d t} - \frac{{\color{red}{\int{\frac{\cos{\left(v \right)}}{2} d v}}}}{32} = \frac{t}{8} - \frac{u}{32} + \int{\frac{\cos{\left(2 t \right)}}{8} d t} - \int{\frac{\cos^{3}{\left(2 t \right)}}{8} d t} - \frac{{\color{red}{\left(\frac{\int{\cos{\left(v \right)} d v}}{2}\right)}}}{32}$$

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

$$\frac{t}{8} - \frac{u}{32} + \int{\frac{\cos{\left(2 t \right)}}{8} d t} - \int{\frac{\cos^{3}{\left(2 t \right)}}{8} d t} - \frac{{\color{red}{\int{\cos{\left(v \right)} d v}}}}{64} = \frac{t}{8} - \frac{u}{32} + \int{\frac{\cos{\left(2 t \right)}}{8} d t} - \int{\frac{\cos^{3}{\left(2 t \right)}}{8} d t} - \frac{{\color{red}{\sin{\left(v \right)}}}}{64}$$

回顧一下 $$$v=2 u$$$：

$$\frac{t}{8} - \frac{u}{32} + \int{\frac{\cos{\left(2 t \right)}}{8} d t} - \int{\frac{\cos^{3}{\left(2 t \right)}}{8} d t} - \frac{\sin{\left({\color{red}{v}} \right)}}{64} = \frac{t}{8} - \frac{u}{32} + \int{\frac{\cos{\left(2 t \right)}}{8} d t} - \int{\frac{\cos^{3}{\left(2 t \right)}}{8} d t} - \frac{\sin{\left({\color{red}{\left(2 u\right)}} \right)}}{64}$$

回顧一下 $$$u=2 t$$$：

$$\frac{t}{8} + \int{\frac{\cos{\left(2 t \right)}}{8} d t} - \int{\frac{\cos^{3}{\left(2 t \right)}}{8} d t} - \frac{\sin{\left(2 {\color{red}{u}} \right)}}{64} - \frac{{\color{red}{u}}}{32} = \frac{t}{8} + \int{\frac{\cos{\left(2 t \right)}}{8} d t} - \int{\frac{\cos^{3}{\left(2 t \right)}}{8} d t} - \frac{\sin{\left(2 {\color{red}{\left(2 t\right)}} \right)}}{64} - \frac{{\color{red}{\left(2 t\right)}}}{32}$$

套用常數倍法則 $$$\int c f{\left(t \right)}\, dt = c \int f{\left(t \right)}\, dt$$$，使用 $$$c=\frac{1}{8}$$$ 與 $$$f{\left(t \right)} = \cos^{3}{\left(2 t \right)}$$$：

$$\frac{t}{16} - \frac{\sin{\left(4 t \right)}}{64} + \int{\frac{\cos{\left(2 t \right)}}{8} d t} - {\color{red}{\int{\frac{\cos^{3}{\left(2 t \right)}}{8} d t}}} = \frac{t}{16} - \frac{\sin{\left(4 t \right)}}{64} + \int{\frac{\cos{\left(2 t \right)}}{8} d t} - {\color{red}{\left(\frac{\int{\cos^{3}{\left(2 t \right)} d t}}{8}\right)}}$$

令 $$$u=2 t$$$。

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

該積分可改寫為

$$\frac{t}{16} - \frac{\sin{\left(4 t \right)}}{64} + \int{\frac{\cos{\left(2 t \right)}}{8} d t} - \frac{{\color{red}{\int{\cos^{3}{\left(2 t \right)} d t}}}}{8} = \frac{t}{16} - \frac{\sin{\left(4 t \right)}}{64} + \int{\frac{\cos{\left(2 t \right)}}{8} d t} - \frac{{\color{red}{\int{\frac{\cos^{3}{\left(u \right)}}{2} d u}}}}{8}$$

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 $$$f{\left(u \right)} = \cos^{3}{\left(u \right)}$$$：

$$\frac{t}{16} - \frac{\sin{\left(4 t \right)}}{64} + \int{\frac{\cos{\left(2 t \right)}}{8} d t} - \frac{{\color{red}{\int{\frac{\cos^{3}{\left(u \right)}}{2} d u}}}}{8} = \frac{t}{16} - \frac{\sin{\left(4 t \right)}}{64} + \int{\frac{\cos{\left(2 t \right)}}{8} d t} - \frac{{\color{red}{\left(\frac{\int{\cos^{3}{\left(u \right)} d u}}{2}\right)}}}{8}$$

先提出一個餘弦，並將其餘部分以正弦表示，使用公式 $$$\cos^2\left(\alpha \right)=-\sin^2\left(\alpha \right)+1$$$，令 $$$\alpha= u $$$:

$$\frac{t}{16} - \frac{\sin{\left(4 t \right)}}{64} + \int{\frac{\cos{\left(2 t \right)}}{8} d t} - \frac{{\color{red}{\int{\cos^{3}{\left(u \right)} d u}}}}{16} = \frac{t}{16} - \frac{\sin{\left(4 t \right)}}{64} + \int{\frac{\cos{\left(2 t \right)}}{8} d t} - \frac{{\color{red}{\int{\left(1 - \sin^{2}{\left(u \right)}\right) \cos{\left(u \right)} d u}}}}{16}$$

令 $$$v=\sin{\left(u \right)}$$$。

則 $$$dv=\left(\sin{\left(u \right)}\right)^{\prime }du = \cos{\left(u \right)} du$$$ (步驟見»)，並可得 $$$\cos{\left(u \right)} du = dv$$$。

該積分可改寫為

$$\frac{t}{16} - \frac{\sin{\left(4 t \right)}}{64} + \int{\frac{\cos{\left(2 t \right)}}{8} d t} - \frac{{\color{red}{\int{\left(1 - \sin^{2}{\left(u \right)}\right) \cos{\left(u \right)} d u}}}}{16} = \frac{t}{16} - \frac{\sin{\left(4 t \right)}}{64} + \int{\frac{\cos{\left(2 t \right)}}{8} d t} - \frac{{\color{red}{\int{\left(1 - v^{2}\right)d v}}}}{16}$$

逐項積分：

$$\frac{t}{16} - \frac{\sin{\left(4 t \right)}}{64} + \int{\frac{\cos{\left(2 t \right)}}{8} d t} - \frac{{\color{red}{\int{\left(1 - v^{2}\right)d v}}}}{16} = \frac{t}{16} - \frac{\sin{\left(4 t \right)}}{64} + \int{\frac{\cos{\left(2 t \right)}}{8} d t} - \frac{{\color{red}{\left(\int{1 d v} - \int{v^{2} d v}\right)}}}{16}$$

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

$$\frac{t}{16} - \frac{\sin{\left(4 t \right)}}{64} + \int{\frac{\cos{\left(2 t \right)}}{8} d t} + \frac{\int{v^{2} d v}}{16} - \frac{{\color{red}{\int{1 d v}}}}{16} = \frac{t}{16} - \frac{\sin{\left(4 t \right)}}{64} + \int{\frac{\cos{\left(2 t \right)}}{8} d t} + \frac{\int{v^{2} d v}}{16} - \frac{{\color{red}{v}}}{16}$$

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

$$\frac{t}{16} - \frac{v}{16} - \frac{\sin{\left(4 t \right)}}{64} + \int{\frac{\cos{\left(2 t \right)}}{8} d t} + \frac{{\color{red}{\int{v^{2} d v}}}}{16}=\frac{t}{16} - \frac{v}{16} - \frac{\sin{\left(4 t \right)}}{64} + \int{\frac{\cos{\left(2 t \right)}}{8} d t} + \frac{{\color{red}{\frac{v^{1 + 2}}{1 + 2}}}}{16}=\frac{t}{16} - \frac{v}{16} - \frac{\sin{\left(4 t \right)}}{64} + \int{\frac{\cos{\left(2 t \right)}}{8} d t} + \frac{{\color{red}{\left(\frac{v^{3}}{3}\right)}}}{16}$$

回顧一下 $$$v=\sin{\left(u \right)}$$$：

$$\frac{t}{16} - \frac{\sin{\left(4 t \right)}}{64} + \int{\frac{\cos{\left(2 t \right)}}{8} d t} - \frac{{\color{red}{v}}}{16} + \frac{{\color{red}{v}}^{3}}{48} = \frac{t}{16} - \frac{\sin{\left(4 t \right)}}{64} + \int{\frac{\cos{\left(2 t \right)}}{8} d t} - \frac{{\color{red}{\sin{\left(u \right)}}}}{16} + \frac{{\color{red}{\sin{\left(u \right)}}}^{3}}{48}$$

回顧一下 $$$u=2 t$$$：

$$\frac{t}{16} - \frac{\sin{\left(4 t \right)}}{64} + \int{\frac{\cos{\left(2 t \right)}}{8} d t} - \frac{\sin{\left({\color{red}{u}} \right)}}{16} + \frac{\sin^{3}{\left({\color{red}{u}} \right)}}{48} = \frac{t}{16} - \frac{\sin{\left(4 t \right)}}{64} + \int{\frac{\cos{\left(2 t \right)}}{8} d t} - \frac{\sin{\left({\color{red}{\left(2 t\right)}} \right)}}{16} + \frac{\sin^{3}{\left({\color{red}{\left(2 t\right)}} \right)}}{48}$$

套用常數倍法則 $$$\int c f{\left(t \right)}\, dt = c \int f{\left(t \right)}\, dt$$$，使用 $$$c=\frac{1}{8}$$$ 與 $$$f{\left(t \right)} = \cos{\left(2 t \right)}$$$：

$$\frac{t}{16} + \frac{\sin^{3}{\left(2 t \right)}}{48} - \frac{\sin{\left(2 t \right)}}{16} - \frac{\sin{\left(4 t \right)}}{64} + {\color{red}{\int{\frac{\cos{\left(2 t \right)}}{8} d t}}} = \frac{t}{16} + \frac{\sin^{3}{\left(2 t \right)}}{48} - \frac{\sin{\left(2 t \right)}}{16} - \frac{\sin{\left(4 t \right)}}{64} + {\color{red}{\left(\frac{\int{\cos{\left(2 t \right)} d t}}{8}\right)}}$$

積分 $$$\int{\cos{\left(2 t \right)} d t}$$$ 已經計算過：

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

因此，

$$\frac{t}{16} + \frac{\sin^{3}{\left(2 t \right)}}{48} - \frac{\sin{\left(2 t \right)}}{16} - \frac{\sin{\left(4 t \right)}}{64} + \frac{{\color{red}{\int{\cos{\left(2 t \right)} d t}}}}{8} = \frac{t}{16} + \frac{\sin^{3}{\left(2 t \right)}}{48} - \frac{\sin{\left(2 t \right)}}{16} - \frac{\sin{\left(4 t \right)}}{64} + \frac{{\color{red}{\left(\frac{\sin{\left(2 t \right)}}{2}\right)}}}{8}$$

因此，

$$\int{\sin^{2}{\left(t \right)} \cos^{4}{\left(t \right)} d t} = \frac{t}{16} + \frac{\sin^{3}{\left(2 t \right)}}{48} - \frac{\sin{\left(4 t \right)}}{64}$$

加上積分常數：

$$\int{\sin^{2}{\left(t \right)} \cos^{4}{\left(t \right)} d t} = \frac{t}{16} + \frac{\sin^{3}{\left(2 t \right)}}{48} - \frac{\sin{\left(4 t \right)}}{64}+C$$

$$$\int \sin^{2}{\left(t \right)} \cos^{4}{\left(t \right)}\, dt = \left(\frac{t}{16} + \frac{\sin^{3}{\left(2 t \right)}}{48} - \frac{\sin{\left(4 t \right)}}{64}\right) + C$$$A