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

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

Integreer termgewijs:

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

Pas de constante-veelvoudregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ toe met $$$c=e^{2}$$$ en $$$f{\left(x \right)} = x$$$:

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

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

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

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

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

Dus,

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

Voeg de integratieconstante toe:

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

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