Integral of $$$- 6 e x^{2} - 2 x$$$

Find $$$\int \left(- 6 e x^{2} - 2 x\right)\, dx$$$.

Integrate term by term:

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

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

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

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

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

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=6 e$$$ and $$$f{\left(x \right)} = x^{2}$$$:

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

Apply the power rule $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=2$$$:

$$- x^{2} - 6 e {\color{red}{\int{x^{2} d x}}}=- x^{2} - 6 e {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}=- x^{2} - 6 e {\color{red}{\left(\frac{x^{3}}{3}\right)}}$$

Therefore,

$$\int{\left(- 6 e x^{2} - 2 x\right)d x} = - 2 e x^{3} - x^{2}$$

Simplify:

$$\int{\left(- 6 e x^{2} - 2 x\right)d x} = x^{2} \left(- 2 e x - 1\right)$$

Add the constant of integration:

$$\int{\left(- 6 e x^{2} - 2 x\right)d x} = x^{2} \left(- 2 e x - 1\right)+C$$

$$$\int \left(- 6 e x^{2} - 2 x\right)\, dx = x^{2} \left(- 2 e x - 1\right) + C$$$A