Integral of $$$2 p - q$$$ with respect to $$$p$$$

Find $$$\int \left(2 p - q\right)\, dp$$$.

Integrate term by term:

$${\color{red}{\int{\left(2 p - q\right)d p}}} = {\color{red}{\left(\int{2 p d p} - \int{q d p}\right)}}$$

Apply the constant rule $$$\int c\, dp = c p$$$ with $$$c=q$$$:

$$\int{2 p d p} - {\color{red}{\int{q d p}}} = \int{2 p d p} - {\color{red}{p q}}$$

Apply the constant multiple rule $$$\int c f{\left(p \right)}\, dp = c \int f{\left(p \right)}\, dp$$$ with $$$c=2$$$ and $$$f{\left(p \right)} = p$$$:

$$- p q + {\color{red}{\int{2 p d p}}} = - p q + {\color{red}{\left(2 \int{p d p}\right)}}$$

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

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

Therefore,

$$\int{\left(2 p - q\right)d p} = p^{2} - p q$$

Simplify:

$$\int{\left(2 p - q\right)d p} = p \left(p - q\right)$$

Add the constant of integration:

$$\int{\left(2 p - q\right)d p} = p \left(p - q\right)+C$$

$$$\int \left(2 p - q\right)\, dp = p \left(p - q\right) + C$$$A