Integral of $$$- 2 x + x e^{3}$$$

Find $$$\int \left(- 2 x + x e^{3}\right)\, dx$$$.

Integrate term by term:

$${\color{red}{\int{\left(- 2 x + x e^{3}\right)d x}}} = {\color{red}{\left(- \int{2 x d x} + \int{x e^{3} d x}\right)}}$$

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

$$\int{x e^{3} d x} - {\color{red}{\int{2 x d x}}} = \int{x e^{3} d x} - {\color{red}{\left(2 \int{x d x}\right)}}$$

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

$$\int{x e^{3} d x} - 2 {\color{red}{\int{x d x}}}=\int{x e^{3} d x} - 2 {\color{red}{\frac{x^{1 + 1}}{1 + 1}}}=\int{x e^{3} d x} - 2 {\color{red}{\left(\frac{x^{2}}{2}\right)}}$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=e^{3}$$$ and $$$f{\left(x \right)} = x$$$:

$$- x^{2} + {\color{red}{\int{x e^{3} d x}}} = - x^{2} + {\color{red}{e^{3} \int{x d x}}}$$

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

$$- x^{2} + e^{3} {\color{red}{\int{x d x}}}=- x^{2} + e^{3} {\color{red}{\frac{x^{1 + 1}}{1 + 1}}}=- x^{2} + e^{3} {\color{red}{\left(\frac{x^{2}}{2}\right)}}$$

Therefore,

$$\int{\left(- 2 x + x e^{3}\right)d x} = - x^{2} + \frac{x^{2} e^{3}}{2}$$

Simplify:

$$\int{\left(- 2 x + x e^{3}\right)d x} = \frac{x^{2} \left(-2 + e^{3}\right)}{2}$$

Add the constant of integration:

$$\int{\left(- 2 x + x e^{3}\right)d x} = \frac{x^{2} \left(-2 + e^{3}\right)}{2}+C$$

$$$\int \left(- 2 x + x e^{3}\right)\, dx = \frac{x^{2} \left(-2 + e^{3}\right)}{2} + C$$$A