Integral of $$$8 x^{9} y^{3}$$$ with respect to $$$x$$$

Find $$$\int 8 x^{9} y^{3}\, dx$$$.

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=8 y^{3}$$$ and $$$f{\left(x \right)} = x^{9}$$$:

$${\color{red}{\int{8 x^{9} y^{3} d x}}} = {\color{red}{\left(8 y^{3} \int{x^{9} d x}\right)}}$$

Apply the power rule $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=9$$$:

$$8 y^{3} {\color{red}{\int{x^{9} d x}}}=8 y^{3} {\color{red}{\frac{x^{1 + 9}}{1 + 9}}}=8 y^{3} {\color{red}{\left(\frac{x^{10}}{10}\right)}}$$

Therefore,

$$\int{8 x^{9} y^{3} d x} = \frac{4 x^{10} y^{3}}{5}$$

Add the constant of integration:

$$\int{8 x^{9} y^{3} d x} = \frac{4 x^{10} y^{3}}{5}+C$$

$$$\int 8 x^{9} y^{3}\, dx = \frac{4 x^{10} y^{3}}{5} + C$$$A