Integral of $$$2 x e^{2}$$$

Find $$$\int 2 x e^{2}\, dx$$$.

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=2 e^{2}$$$ and $$$f{\left(x \right)} = x$$$:

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

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

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

Therefore,

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

Add the constant of integration:

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

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