Integral of $$$x e^{2} - 1$$$

Find $$$\int \left(x e^{2} - 1\right)\, dx$$$.

Integrate term by term:

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

Apply the constant rule $$$\int c\, dx = c x$$$ with $$$c=1$$$:

$$\int{x e^{2} d x} - {\color{red}{\int{1 d x}}} = \int{x e^{2} d x} - {\color{red}{x}}$$

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

$$- x + {\color{red}{\int{x e^{2} d x}}} = - x + {\color{red}{e^{2} \int{x d x}}}$$

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

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

Therefore,

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

Simplify:

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

Add the constant of integration:

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

$$$\int \left(x e^{2} - 1\right)\, dx = \frac{x \left(x e^{2} - 2\right)}{2} + C$$$A