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

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

Integrate term by term:

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

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

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

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

$$\frac{x^{2}}{2} + {\color{red}{\int{y^{2} d x}}} = \frac{x^{2}}{2} + {\color{red}{x y^{2}}}$$

Therefore,

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

Simplify:

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

Add the constant of integration:

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

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