Integral of $$$- x \left(1 - 2 x\right) + 1$$$

Find $$$\int \left(- x \left(1 - 2 x\right) + 1\right)\, dx$$$.

Integrate term by term:

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

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

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

Expand the expression:

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

Integrate term by term:

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

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

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

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

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

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

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

Therefore,

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

Simplify:

$$\int{\left(- x \left(1 - 2 x\right) + 1\right)d x} = \frac{x \left(4 x^{2} - 3 x + 6\right)}{6}$$

Add the constant of integration:

$$\int{\left(- x \left(1 - 2 x\right) + 1\right)d x} = \frac{x \left(4 x^{2} - 3 x + 6\right)}{6}+C$$

$$$\int \left(- x \left(1 - 2 x\right) + 1\right)\, dx = \frac{x \left(4 x^{2} - 3 x + 6\right)}{6} + C$$$A