$$$\sin^{9}{\left(\theta \right)} \cos{\left(\theta \right)}$$$ 的積分

求$$$\int \sin^{9}{\left(\theta \right)} \cos{\left(\theta \right)}\, d\theta$$$。

令 $$$u=\sin{\left(\theta \right)}$$$。

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

因此，

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

套用冪次法則 $$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$，以 $$$n=9$$$：

$${\color{red}{\int{u^{9} d u}}}={\color{red}{\frac{u^{1 + 9}}{1 + 9}}}={\color{red}{\left(\frac{u^{10}}{10}\right)}}$$

回顧一下 $$$u=\sin{\left(\theta \right)}$$$：

$$\frac{{\color{red}{u}}^{10}}{10} = \frac{{\color{red}{\sin{\left(\theta \right)}}}^{10}}{10}$$

因此，

$$\int{\sin^{9}{\left(\theta \right)} \cos{\left(\theta \right)} d \theta} = \frac{\sin^{10}{\left(\theta \right)}}{10}$$

加上積分常數：

$$\int{\sin^{9}{\left(\theta \right)} \cos{\left(\theta \right)} d \theta} = \frac{\sin^{10}{\left(\theta \right)}}{10}+C$$

$$$\int \sin^{9}{\left(\theta \right)} \cos{\left(\theta \right)}\, d\theta = \frac{\sin^{10}{\left(\theta \right)}}{10} + C$$$A