Integral of $$$w - \frac{3}{2}$$$

Find $$$\int \left(w - \frac{3}{2}\right)\, dw$$$.

Integrate term by term:

$${\color{red}{\int{\left(w - \frac{3}{2}\right)d w}}} = {\color{red}{\left(- \int{\frac{3}{2} d w} + \int{w d w}\right)}}$$

Apply the constant rule $$$\int c\, dw = c w$$$ with $$$c=\frac{3}{2}$$$:

$$\int{w d w} - {\color{red}{\int{\frac{3}{2} d w}}} = \int{w d w} - {\color{red}{\left(\frac{3 w}{2}\right)}}$$

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

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

Therefore,

$$\int{\left(w - \frac{3}{2}\right)d w} = \frac{w^{2}}{2} - \frac{3 w}{2}$$

Simplify:

$$\int{\left(w - \frac{3}{2}\right)d w} = \frac{w \left(w - 3\right)}{2}$$

Add the constant of integration:

$$\int{\left(w - \frac{3}{2}\right)d w} = \frac{w \left(w - 3\right)}{2}+C$$

$$$\int \left(w - \frac{3}{2}\right)\, dw = \frac{w \left(w - 3\right)}{2} + C$$$A