Integral of $$$x e^{6}$$$

Find $$$\int x e^{6}\, dx$$$.

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=e^{6}$$$ and $$$f{\left(x \right)} = x$$$:

$${\color{red}{\int{x e^{6} d x}}} = {\color{red}{e^{6} \int{x d x}}}$$

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

$$e^{6} {\color{red}{\int{x d x}}}=e^{6} {\color{red}{\frac{x^{1 + 1}}{1 + 1}}}=e^{6} {\color{red}{\left(\frac{x^{2}}{2}\right)}}$$

Therefore,

$$\int{x e^{6} d x} = \frac{x^{2} e^{6}}{2}$$

Add the constant of integration:

$$\int{x e^{6} d x} = \frac{x^{2} e^{6}}{2}+C$$

$$$\int x e^{6}\, dx = \frac{x^{2} e^{6}}{2} + C$$$A