Integraal van $$$e^{\frac{t}{50}}$$$

Bepaal $$$\int e^{\frac{t}{50}}\, dt$$$.

Zij $$$u=\frac{t}{50}$$$.

Dan $$$du=\left(\frac{t}{50}\right)^{\prime }dt = \frac{dt}{50}$$$ (de stappen zijn te zien »), en dan geldt dat $$$dt = 50 du$$$.

Dus,

$${\color{red}{\int{e^{\frac{t}{50}} d t}}} = {\color{red}{\int{50 e^{u} d u}}}$$

Pas de constante-veelvoudregel $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ toe met $$$c=50$$$ en $$$f{\left(u \right)} = e^{u}$$$:

$${\color{red}{\int{50 e^{u} d u}}} = {\color{red}{\left(50 \int{e^{u} d u}\right)}}$$

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

$$50 {\color{red}{\int{e^{u} d u}}} = 50 {\color{red}{e^{u}}}$$

We herinneren eraan dat $$$u=\frac{t}{50}$$$:

$$50 e^{{\color{red}{u}}} = 50 e^{{\color{red}{\left(\frac{t}{50}\right)}}}$$

Dus,

$$\int{e^{\frac{t}{50}} d t} = 50 e^{\frac{t}{50}}$$

Voeg de integratieconstante toe:

$$\int{e^{\frac{t}{50}} d t} = 50 e^{\frac{t}{50}}+C$$

$$$\int e^{\frac{t}{50}}\, dt = 50 e^{\frac{t}{50}} + C$$$A