Integral of $$$_1 x^{11}$$$ with respect to $$$x$$$

Find $$$\int _1 x^{11}\, dx$$$.

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=_1$$$ and $$$f{\left(x \right)} = x^{11}$$$:

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

Apply the power rule $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=11$$$:

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

Therefore,

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

Add the constant of integration:

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

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