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

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

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 $$$\sin{\left(x \right)} dx = - du$$$。

該積分可改寫為

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

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

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

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 $$$u=\cos{\left(x \right)}$$$：

$$- \frac{{\color{red}{u}}^{3}}{3} = - \frac{{\color{red}{\cos{\left(x \right)}}}^{3}}{3}$$

因此，

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

加上積分常數：

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

$$$\int \sin{\left(x \right)} \cos^{2}{\left(x \right)}\, dx = - \frac{\cos^{3}{\left(x \right)}}{3} + C$$$A