Integraal van $$$e^{4 \theta}$$$

Bepaal $$$\int e^{4 \theta}\, d\theta$$$.

Zij $$$u=4 \theta$$$.

Dan $$$du=\left(4 \theta\right)^{\prime }d\theta = 4 d\theta$$$ (de stappen zijn te zien »), en dan geldt dat $$$d\theta = \frac{du}{4}$$$.

De integraal kan worden herschreven als

$${\color{red}{\int{e^{4 \theta} d \theta}}} = {\color{red}{\int{\frac{e^{u}}{4} d u}}}$$

Pas de constante-veelvoudregel $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ toe met $$$c=\frac{1}{4}$$$ en $$$f{\left(u \right)} = e^{u}$$$:

$${\color{red}{\int{\frac{e^{u}}{4} d u}}} = {\color{red}{\left(\frac{\int{e^{u} d u}}{4}\right)}}$$

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

$$\frac{{\color{red}{\int{e^{u} d u}}}}{4} = \frac{{\color{red}{e^{u}}}}{4}$$

We herinneren eraan dat $$$u=4 \theta$$$:

$$\frac{e^{{\color{red}{u}}}}{4} = \frac{e^{{\color{red}{\left(4 \theta\right)}}}}{4}$$

Dus,

$$\int{e^{4 \theta} d \theta} = \frac{e^{4 \theta}}{4}$$

Voeg de integratieconstante toe:

$$\int{e^{4 \theta} d \theta} = \frac{e^{4 \theta}}{4}+C$$

$$$\int e^{4 \theta}\, d\theta = \frac{e^{4 \theta}}{4} + C$$$A