Integral von $$$e^{\cos{\left(y \right)}} \sin{\left(y \right)}$$$

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

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

Dann $$$du=\left(\cos{\left(y \right)}\right)^{\prime }dy = - \sin{\left(y \right)} dy$$$ (die Schritte sind » zu sehen), und es gilt $$$\sin{\left(y \right)} dy = - du$$$.

Also,

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

Wende die Konstantenfaktorregel $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ mit $$$c=-1$$$ und $$$f{\left(u \right)} = e^{u}$$$ an:

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

Das Integral der Exponentialfunktion lautet $$$\int{e^{u} d u} = e^{u}$$$:

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

Zur Erinnerung: $$$u=\cos{\left(y \right)}$$$:

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

Daher,

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

Fügen Sie die Integrationskonstante hinzu:

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

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