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

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

Expand the expression:

$${\color{red}{\int{\left(x^{2} + y^{2}\right)^{2} d x}}} = {\color{red}{\int{\left(x^{4} + 2 x^{2} y^{2} + y^{4}\right)d x}}}$$

Integrate term by term:

$${\color{red}{\int{\left(x^{4} + 2 x^{2} y^{2} + y^{4}\right)d x}}} = {\color{red}{\left(\int{x^{4} d x} + \int{y^{4} d x} + \int{2 x^{2} y^{2} d x}\right)}}$$

Apply the power rule $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=4$$$:

$$\int{y^{4} d x} + \int{2 x^{2} y^{2} d x} + {\color{red}{\int{x^{4} d x}}}=\int{y^{4} d x} + \int{2 x^{2} y^{2} d x} + {\color{red}{\frac{x^{1 + 4}}{1 + 4}}}=\int{y^{4} d x} + \int{2 x^{2} y^{2} d x} + {\color{red}{\left(\frac{x^{5}}{5}\right)}}$$

Apply the constant rule $$$\int c\, dx = c x$$$ with $$$c=y^{4}$$$:

$$\frac{x^{5}}{5} + \int{2 x^{2} y^{2} d x} + {\color{red}{\int{y^{4} d x}}} = \frac{x^{5}}{5} + \int{2 x^{2} y^{2} d x} + {\color{red}{x y^{4}}}$$

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

$$\frac{x^{5}}{5} + x y^{4} + {\color{red}{\int{2 x^{2} y^{2} d x}}} = \frac{x^{5}}{5} + x y^{4} + {\color{red}{\left(2 y^{2} \int{x^{2} d x}\right)}}$$

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

$$\frac{x^{5}}{5} + x y^{4} + 2 y^{2} {\color{red}{\int{x^{2} d x}}}=\frac{x^{5}}{5} + x y^{4} + 2 y^{2} {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}=\frac{x^{5}}{5} + x y^{4} + 2 y^{2} {\color{red}{\left(\frac{x^{3}}{3}\right)}}$$

Therefore,

$$\int{\left(x^{2} + y^{2}\right)^{2} d x} = \frac{x^{5}}{5} + \frac{2 x^{3} y^{2}}{3} + x y^{4}$$

Simplify:

$$\int{\left(x^{2} + y^{2}\right)^{2} d x} = x \left(\frac{x^{4}}{5} + \frac{2 x^{2} y^{2}}{3} + y^{4}\right)$$

Add the constant of integration:

$$\int{\left(x^{2} + y^{2}\right)^{2} d x} = x \left(\frac{x^{4}}{5} + \frac{2 x^{2} y^{2}}{3} + y^{4}\right)+C$$

$$$\int \left(x^{2} + y^{2}\right)^{2}\, dx = x \left(\frac{x^{4}}{5} + \frac{2 x^{2} y^{2}}{3} + y^{4}\right) + C$$$A