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

求$$$\int \sin{\left(x \right)} \cos^{2}{\left(\cos{\left(x \right)} \right)}\, dx$$$。

令 $$$u=\cos{\left(x \right)}$$$。

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

所以，

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

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

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

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

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

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

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

逐項積分：

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

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

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

令 $$$v=2 u$$$。

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

因此，

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

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

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

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

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

回顧一下 $$$v=2 u$$$：

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

回顧一下 $$$u=\cos{\left(x \right)}$$$：

$$- \frac{\sin{\left(2 {\color{red}{u}} \right)}}{4} - \frac{{\color{red}{u}}}{2} = - \frac{\sin{\left(2 {\color{red}{\cos{\left(x \right)}}} \right)}}{4} - \frac{{\color{red}{\cos{\left(x \right)}}}}{2}$$

因此，

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

加上積分常數：

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

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