Integraal van $$$e^{\frac{x}{5}}$$$

Bepaal $$$\int e^{\frac{x}{5}}\, dx$$$.

Zij $$$u=\frac{x}{5}$$$.

Dan $$$du=\left(\frac{x}{5}\right)^{\prime }dx = \frac{dx}{5}$$$ (de stappen zijn te zien »), en dan geldt dat $$$dx = 5 du$$$.

Dus,

$${\color{red}{\int{e^{\frac{x}{5}} d x}}} = {\color{red}{\int{5 e^{u} d u}}}$$

Pas de constante-veelvoudregel $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ toe met $$$c=5$$$ en $$$f{\left(u \right)} = e^{u}$$$:

$${\color{red}{\int{5 e^{u} d u}}} = {\color{red}{\left(5 \int{e^{u} d u}\right)}}$$

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

$$5 {\color{red}{\int{e^{u} d u}}} = 5 {\color{red}{e^{u}}}$$

We herinneren eraan dat $$$u=\frac{x}{5}$$$:

$$5 e^{{\color{red}{u}}} = 5 e^{{\color{red}{\left(\frac{x}{5}\right)}}}$$

Dus,

$$\int{e^{\frac{x}{5}} d x} = 5 e^{\frac{x}{5}}$$

Voeg de integratieconstante toe:

$$\int{e^{\frac{x}{5}} d x} = 5 e^{\frac{x}{5}}+C$$

$$$\int e^{\frac{x}{5}}\, dx = 5 e^{\frac{x}{5}} + C$$$A