Integral of $$$\frac{3 t^{2}}{1000} - \frac{t}{125}$$$

Find $$$\int \left(\frac{3 t^{2}}{1000} - \frac{t}{125}\right)\, dt$$$.

Integrate term by term:

$${\color{red}{\int{\left(\frac{3 t^{2}}{1000} - \frac{t}{125}\right)d t}}} = {\color{red}{\left(- \int{\frac{t}{125} d t} + \int{\frac{3 t^{2}}{1000} d t}\right)}}$$

Apply the constant multiple rule $$$\int c f{\left(t \right)}\, dt = c \int f{\left(t \right)}\, dt$$$ with $$$c=\frac{1}{125}$$$ and $$$f{\left(t \right)} = t$$$:

$$\int{\frac{3 t^{2}}{1000} d t} - {\color{red}{\int{\frac{t}{125} d t}}} = \int{\frac{3 t^{2}}{1000} d t} - {\color{red}{\left(\frac{\int{t d t}}{125}\right)}}$$

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

$$\int{\frac{3 t^{2}}{1000} d t} - \frac{{\color{red}{\int{t d t}}}}{125}=\int{\frac{3 t^{2}}{1000} d t} - \frac{{\color{red}{\frac{t^{1 + 1}}{1 + 1}}}}{125}=\int{\frac{3 t^{2}}{1000} d t} - \frac{{\color{red}{\left(\frac{t^{2}}{2}\right)}}}{125}$$

Apply the constant multiple rule $$$\int c f{\left(t \right)}\, dt = c \int f{\left(t \right)}\, dt$$$ with $$$c=\frac{3}{1000}$$$ and $$$f{\left(t \right)} = t^{2}$$$:

$$- \frac{t^{2}}{250} + {\color{red}{\int{\frac{3 t^{2}}{1000} d t}}} = - \frac{t^{2}}{250} + {\color{red}{\left(\frac{3 \int{t^{2} d t}}{1000}\right)}}$$

Apply the power rule $$$\int t^{n}\, dt = \frac{t^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=2$$$:

$$- \frac{t^{2}}{250} + \frac{3 {\color{red}{\int{t^{2} d t}}}}{1000}=- \frac{t^{2}}{250} + \frac{3 {\color{red}{\frac{t^{1 + 2}}{1 + 2}}}}{1000}=- \frac{t^{2}}{250} + \frac{3 {\color{red}{\left(\frac{t^{3}}{3}\right)}}}{1000}$$

Therefore,

$$\int{\left(\frac{3 t^{2}}{1000} - \frac{t}{125}\right)d t} = \frac{t^{3}}{1000} - \frac{t^{2}}{250}$$

Simplify:

$$\int{\left(\frac{3 t^{2}}{1000} - \frac{t}{125}\right)d t} = \frac{t^{2} \left(t - 4\right)}{1000}$$

Add the constant of integration:

$$\int{\left(\frac{3 t^{2}}{1000} - \frac{t}{125}\right)d t} = \frac{t^{2} \left(t - 4\right)}{1000}+C$$

$$$\int \left(\frac{3 t^{2}}{1000} - \frac{t}{125}\right)\, dt = \frac{t^{2} \left(t - 4\right)}{1000} + C$$$A