Integral of $$$- 2 f x$$$ with respect to $$$x$$$

Find $$$\int \left(- 2 f x\right)\, dx$$$.

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

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

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

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

Therefore,

$$\int{\left(- 2 f x\right)d x} = - f x^{2}$$

Add the constant of integration:

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

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