Integral of $$$25 i d n t x^{7}$$$ with respect to $$$x$$$

Find $$$\int 25 i d n t x^{7}\, dx$$$.

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=25 i d n t$$$ and $$$f{\left(x \right)} = x^{7}$$$:

$${\color{red}{\int{25 i d n t x^{7} d x}}} = {\color{red}{\left(25 i d n t \int{x^{7} d x}\right)}}$$

Apply the power rule $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=7$$$:

$$25 i d n t {\color{red}{\int{x^{7} d x}}}=25 i d n t {\color{red}{\frac{x^{1 + 7}}{1 + 7}}}=25 i d n t {\color{red}{\left(\frac{x^{8}}{8}\right)}}$$

Therefore,

$$\int{25 i d n t x^{7} d x} = \frac{25 i d n t x^{8}}{8}$$

Add the constant of integration:

$$\int{25 i d n t x^{7} d x} = \frac{25 i d n t x^{8}}{8}+C$$

$$$\int 25 i d n t x^{7}\, dx = \frac{25 i d n t x^{8}}{8} + C$$$A