$$$\sin^{129}{\left(x \right)} \cos{\left(x \right)}$$$ 的積分

求$$$\int \sin^{129}{\left(x \right)} \cos{\left(x \right)}\, dx$$$。

令 $$$u=\sin{\left(x \right)}$$$。

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

所以，

$${\color{red}{\int{\sin^{129}{\left(x \right)} \cos{\left(x \right)} d x}}} = {\color{red}{\int{u^{129} d u}}}$$

套用冪次法則 $$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$，以 $$$n=129$$$：

$${\color{red}{\int{u^{129} d u}}}={\color{red}{\frac{u^{1 + 129}}{1 + 129}}}={\color{red}{\left(\frac{u^{130}}{130}\right)}}$$

回顧一下 $$$u=\sin{\left(x \right)}$$$：

$$\frac{{\color{red}{u}}^{130}}{130} = \frac{{\color{red}{\sin{\left(x \right)}}}^{130}}{130}$$

因此，

$$\int{\sin^{129}{\left(x \right)} \cos{\left(x \right)} d x} = \frac{\sin^{130}{\left(x \right)}}{130}$$

加上積分常數：

$$\int{\sin^{129}{\left(x \right)} \cos{\left(x \right)} d x} = \frac{\sin^{130}{\left(x \right)}}{130}+C$$

$$$\int \sin^{129}{\left(x \right)} \cos{\left(x \right)}\, dx = \frac{\sin^{130}{\left(x \right)}}{130} + C$$$A