Integraal van $$$2 p - q$$$ met betrekking tot $$$p$$$

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

Integreer termgewijs:

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

Pas de constantenregel $$$\int c\, dp = c p$$$ toe met $$$c=q$$$:

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

Pas de constante-veelvoudregel $$$\int c f{\left(p \right)}\, dp = c \int f{\left(p \right)}\, dp$$$ toe met $$$c=2$$$ en $$$f{\left(p \right)} = p$$$:

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

Pas de machtsregel $$$\int p^{n}\, dp = \frac{p^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ toe met $$$n=1$$$:

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

Dus,

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

Vereenvoudig:

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

Voeg de integratieconstante toe:

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

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