Integral of $$$8 a^{8} w^{8}$$$ with respect to $$$a$$$

Find $$$\int 8 a^{8} w^{8}\, da$$$.

Apply the constant multiple rule $$$\int c f{\left(a \right)}\, da = c \int f{\left(a \right)}\, da$$$ with $$$c=8 w^{8}$$$ and $$$f{\left(a \right)} = a^{8}$$$:

$${\color{red}{\int{8 a^{8} w^{8} d a}}} = {\color{red}{\left(8 w^{8} \int{a^{8} d a}\right)}}$$

Apply the power rule $$$\int a^{n}\, da = \frac{a^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=8$$$:

$$8 w^{8} {\color{red}{\int{a^{8} d a}}}=8 w^{8} {\color{red}{\frac{a^{1 + 8}}{1 + 8}}}=8 w^{8} {\color{red}{\left(\frac{a^{9}}{9}\right)}}$$

Therefore,

$$\int{8 a^{8} w^{8} d a} = \frac{8 a^{9} w^{8}}{9}$$

Add the constant of integration:

$$\int{8 a^{8} w^{8} d a} = \frac{8 a^{9} w^{8}}{9}+C$$

$$$\int 8 a^{8} w^{8}\, da = \frac{8 a^{9} w^{8}}{9} + C$$$A