Integraal van $$$e^{x} + 2$$$

Bepaal $$$\int \left(e^{x} + 2\right)\, dx$$$.

Integreer termgewijs:

$${\color{red}{\int{\left(e^{x} + 2\right)d x}}} = {\color{red}{\left(\int{2 d x} + \int{e^{x} d x}\right)}}$$

Pas de constantenregel $$$\int c\, dx = c x$$$ toe met $$$c=2$$$:

$$\int{e^{x} d x} + {\color{red}{\int{2 d x}}} = \int{e^{x} d x} + {\color{red}{\left(2 x\right)}}$$

De integraal van de exponentiële functie is $$$\int{e^{x} d x} = e^{x}$$$:

$$2 x + {\color{red}{\int{e^{x} d x}}} = 2 x + {\color{red}{e^{x}}}$$

Dus,

$$\int{\left(e^{x} + 2\right)d x} = 2 x + e^{x}$$

Voeg de integratieconstante toe:

$$\int{\left(e^{x} + 2\right)d x} = 2 x + e^{x}+C$$

$$$\int \left(e^{x} + 2\right)\, dx = \left(2 x + e^{x}\right) + C$$$A