$$$_1 x^{11}$$$ 對 $$$x$$$ 的積分

求$$$\int _1 x^{11}\, dx$$$。

套用常數倍法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$，使用 $$$c=_1$$$ 與 $$$f{\left(x \right)} = x^{11}$$$：

$${\color{red}{\int{_1 x^{11} d x}}} = {\color{red}{_1 \int{x^{11} d x}}}$$

套用冪次法則 $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$，以 $$$n=11$$$：

$$_1 {\color{red}{\int{x^{11} d x}}}=_1 {\color{red}{\frac{x^{1 + 11}}{1 + 11}}}=_1 {\color{red}{\left(\frac{x^{12}}{12}\right)}}$$

因此，

$$\int{_1 x^{11} d x} = \frac{_1 x^{12}}{12}$$

加上積分常數：

$$\int{_1 x^{11} d x} = \frac{_1 x^{12}}{12}+C$$

$$$\int _1 x^{11}\, dx = \frac{_1 x^{12}}{12} + C$$$A