Integraal van $$$\frac{e^{\frac{x}{200}}}{2}$$$

Bepaal $$$\int \frac{e^{\frac{x}{200}}}{2}\, dx$$$.

Pas de constante-veelvoudregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ toe met $$$c=\frac{1}{2}$$$ en $$$f{\left(x \right)} = e^{\frac{x}{200}}$$$:

$${\color{red}{\int{\frac{e^{\frac{x}{200}}}{2} d x}}} = {\color{red}{\left(\frac{\int{e^{\frac{x}{200}} d x}}{2}\right)}}$$

Zij $$$u=\frac{x}{200}$$$.

Dan $$$du=\left(\frac{x}{200}\right)^{\prime }dx = \frac{dx}{200}$$$ (de stappen zijn te zien »), en dan geldt dat $$$dx = 200 du$$$.

De integraal kan worden herschreven als

$$\frac{{\color{red}{\int{e^{\frac{x}{200}} d x}}}}{2} = \frac{{\color{red}{\int{200 e^{u} d u}}}}{2}$$

Pas de constante-veelvoudregel $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ toe met $$$c=200$$$ en $$$f{\left(u \right)} = e^{u}$$$:

$$\frac{{\color{red}{\int{200 e^{u} d u}}}}{2} = \frac{{\color{red}{\left(200 \int{e^{u} d u}\right)}}}{2}$$

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

$$100 {\color{red}{\int{e^{u} d u}}} = 100 {\color{red}{e^{u}}}$$

We herinneren eraan dat $$$u=\frac{x}{200}$$$:

$$100 e^{{\color{red}{u}}} = 100 e^{{\color{red}{\left(\frac{x}{200}\right)}}}$$

Dus,

$$\int{\frac{e^{\frac{x}{200}}}{2} d x} = 100 e^{\frac{x}{200}}$$

Voeg de integratieconstante toe:

$$\int{\frac{e^{\frac{x}{200}}}{2} d x} = 100 e^{\frac{x}{200}}+C$$

$$$\int \frac{e^{\frac{x}{200}}}{2}\, dx = 100 e^{\frac{x}{200}} + C$$$A