Integral of $$$5880 i n t^{8} - 1$$$ with respect to $$$t$$$

Find $$$\int \left(5880 i n t^{8} - 1\right)\, dt$$$.

Integrate term by term:

$${\color{red}{\int{\left(5880 i n t^{8} - 1\right)d t}}} = {\color{red}{\left(- \int{1 d t} + \int{5880 i n t^{8} d t}\right)}}$$

Apply the constant rule $$$\int c\, dt = c t$$$ with $$$c=1$$$:

$$\int{5880 i n t^{8} d t} - {\color{red}{\int{1 d t}}} = \int{5880 i n t^{8} d t} - {\color{red}{t}}$$

Apply the constant multiple rule $$$\int c f{\left(t \right)}\, dt = c \int f{\left(t \right)}\, dt$$$ with $$$c=5880 i n$$$ and $$$f{\left(t \right)} = t^{8}$$$:

$$- t + {\color{red}{\int{5880 i n t^{8} d t}}} = - t + {\color{red}{\left(5880 i n \int{t^{8} d t}\right)}}$$

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

$$5880 i n {\color{red}{\int{t^{8} d t}}} - t=5880 i n {\color{red}{\frac{t^{1 + 8}}{1 + 8}}} - t=5880 i n {\color{red}{\left(\frac{t^{9}}{9}\right)}} - t$$

Therefore,

$$\int{\left(5880 i n t^{8} - 1\right)d t} = \frac{1960 i n t^{9}}{3} - t$$

Add the constant of integration:

$$\int{\left(5880 i n t^{8} - 1\right)d t} = \frac{1960 i n t^{9}}{3} - t+C$$

$$$\int \left(5880 i n t^{8} - 1\right)\, dt = \left(\frac{1960 i n t^{9}}{3} - t\right) + C$$$A