Integral of $$$x^{2} - 2 y$$$ with respect to $$$x$$$

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

Integrate term by term:

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

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

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

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

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

Therefore,

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

Simplify:

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

Add the constant of integration:

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

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