Integral of $$$r \left(- 18 x + 18 y^{2}\right)$$$ with respect to $$$x$$$

Find $$$\int r \left(- 18 x + 18 y^{2}\right)\, dx$$$.

Simplify the integrand:

$${\color{red}{\int{r \left(- 18 x + 18 y^{2}\right) d x}}} = {\color{red}{\int{18 r \left(- x + y^{2}\right) d x}}}$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=18 r$$$ and $$$f{\left(x \right)} = - x + y^{2}$$$:

$${\color{red}{\int{18 r \left(- x + y^{2}\right) d x}}} = {\color{red}{\left(18 r \int{\left(- x + y^{2}\right)d x}\right)}}$$

Integrate term by term:

$$18 r {\color{red}{\int{\left(- x + y^{2}\right)d x}}} = 18 r {\color{red}{\left(- \int{x d x} + \int{y^{2} d x}\right)}}$$

Apply the constant rule $$$\int c\, dx = c x$$$ with $$$c=y^{2}$$$:

$$18 r \left(- \int{x d x} + {\color{red}{\int{y^{2} d x}}}\right) = 18 r \left(- \int{x d x} + {\color{red}{x y^{2}}}\right)$$

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

$$18 r \left(x y^{2} - {\color{red}{\int{x d x}}}\right)=18 r \left(x y^{2} - {\color{red}{\frac{x^{1 + 1}}{1 + 1}}}\right)=18 r \left(x y^{2} - {\color{red}{\left(\frac{x^{2}}{2}\right)}}\right)$$

Therefore,

$$\int{r \left(- 18 x + 18 y^{2}\right) d x} = 18 r \left(- \frac{x^{2}}{2} + x y^{2}\right)$$

Simplify:

$$\int{r \left(- 18 x + 18 y^{2}\right) d x} = 9 r x \left(- x + 2 y^{2}\right)$$

Add the constant of integration:

$$\int{r \left(- 18 x + 18 y^{2}\right) d x} = 9 r x \left(- x + 2 y^{2}\right)+C$$

$$$\int r \left(- 18 x + 18 y^{2}\right)\, dx = 9 r x \left(- x + 2 y^{2}\right) + C$$$A