Integraal van $$$e^{\cos{\left(y \right)}} \sin{\left(y \right)}$$$

Bepaal $$$\int e^{\cos{\left(y \right)}} \sin{\left(y \right)}\, dy$$$.

Zij $$$u=\cos{\left(y \right)}$$$.

Dan $$$du=\left(\cos{\left(y \right)}\right)^{\prime }dy = - \sin{\left(y \right)} dy$$$ (de stappen zijn te zien »), en dan geldt dat $$$\sin{\left(y \right)} dy = - du$$$.

De integraal wordt

$${\color{red}{\int{e^{\cos{\left(y \right)}} \sin{\left(y \right)} d y}}} = {\color{red}{\int{\left(- e^{u}\right)d u}}}$$

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

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

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

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

We herinneren eraan dat $$$u=\cos{\left(y \right)}$$$:

$$- e^{{\color{red}{u}}} = - e^{{\color{red}{\cos{\left(y \right)}}}}$$

Dus,

$$\int{e^{\cos{\left(y \right)}} \sin{\left(y \right)} d y} = - e^{\cos{\left(y \right)}}$$

Voeg de integratieconstante toe:

$$\int{e^{\cos{\left(y \right)}} \sin{\left(y \right)} d y} = - e^{\cos{\left(y \right)}}+C$$

$$$\int e^{\cos{\left(y \right)}} \sin{\left(y \right)}\, dy = - e^{\cos{\left(y \right)}} + C$$$A