Integral of $$$y^{23} \left(x + y\right)$$$ with respect to $$$x$$$

Find $$$\int y^{23} \left(x + y\right)\, dx$$$.

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=y^{23}$$$ and $$$f{\left(x \right)} = x + y$$$:

$${\color{red}{\int{y^{23} \left(x + y\right) d x}}} = {\color{red}{y^{23} \int{\left(x + y\right)d x}}}$$

Integrate term by term:

$$y^{23} {\color{red}{\int{\left(x + y\right)d x}}} = y^{23} {\color{red}{\left(\int{x d x} + \int{y d x}\right)}}$$

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

$$y^{23} \left(\int{y d x} + {\color{red}{\int{x d x}}}\right)=y^{23} \left(\int{y d x} + {\color{red}{\frac{x^{1 + 1}}{1 + 1}}}\right)=y^{23} \left(\int{y d x} + {\color{red}{\left(\frac{x^{2}}{2}\right)}}\right)$$

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

$$y^{23} \left(\frac{x^{2}}{2} + {\color{red}{\int{y d x}}}\right) = y^{23} \left(\frac{x^{2}}{2} + {\color{red}{x y}}\right)$$

Therefore,

$$\int{y^{23} \left(x + y\right) d x} = y^{23} \left(\frac{x^{2}}{2} + x y\right)$$

Simplify:

$$\int{y^{23} \left(x + y\right) d x} = \frac{x y^{23} \left(x + 2 y\right)}{2}$$

Add the constant of integration:

$$\int{y^{23} \left(x + y\right) d x} = \frac{x y^{23} \left(x + 2 y\right)}{2}+C$$

$$$\int y^{23} \left(x + y\right)\, dx = \frac{x y^{23} \left(x + 2 y\right)}{2} + C$$$A