Integral von $$$5 e^{5 s} \sin{\left(e^{5 s} \right)}$$$

Bestimme $$$\int 5 e^{5 s} \sin{\left(e^{5 s} \right)}\, ds$$$.

Wende die Konstantenfaktorregel $$$\int c f{\left(s \right)}\, ds = c \int f{\left(s \right)}\, ds$$$ mit $$$c=5$$$ und $$$f{\left(s \right)} = e^{5 s} \sin{\left(e^{5 s} \right)}$$$ an:

$${\color{red}{\int{5 e^{5 s} \sin{\left(e^{5 s} \right)} d s}}} = {\color{red}{\left(5 \int{e^{5 s} \sin{\left(e^{5 s} \right)} d s}\right)}}$$

Sei $$$u=5 s$$$.

Dann $$$du=\left(5 s\right)^{\prime }ds = 5 ds$$$ (die Schritte sind » zu sehen), und es gilt $$$ds = \frac{du}{5}$$$.

Daher,

$$5 {\color{red}{\int{e^{5 s} \sin{\left(e^{5 s} \right)} d s}}} = 5 {\color{red}{\int{\frac{e^{u} \sin{\left(e^{u} \right)}}{5} d u}}}$$

Wende die Konstantenfaktorregel $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ mit $$$c=\frac{1}{5}$$$ und $$$f{\left(u \right)} = e^{u} \sin{\left(e^{u} \right)}$$$ an:

$$5 {\color{red}{\int{\frac{e^{u} \sin{\left(e^{u} \right)}}{5} d u}}} = 5 {\color{red}{\left(\frac{\int{e^{u} \sin{\left(e^{u} \right)} d u}}{5}\right)}}$$

Sei $$$v=e^{u}$$$.

Dann $$$dv=\left(e^{u}\right)^{\prime }du = e^{u} du$$$ (die Schritte sind » zu sehen), und es gilt $$$e^{u} du = dv$$$.

Das Integral wird zu

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

Das Integral des Sinus lautet $$$\int{\sin{\left(v \right)} d v} = - \cos{\left(v \right)}$$$:

$${\color{red}{\int{\sin{\left(v \right)} d v}}} = {\color{red}{\left(- \cos{\left(v \right)}\right)}}$$

Zur Erinnerung: $$$v=e^{u}$$$:

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

Zur Erinnerung: $$$u=5 s$$$:

$$- \cos{\left(e^{{\color{red}{u}}} \right)} = - \cos{\left(e^{{\color{red}{\left(5 s\right)}}} \right)}$$

Daher,

$$\int{5 e^{5 s} \sin{\left(e^{5 s} \right)} d s} = - \cos{\left(e^{5 s} \right)}$$

Fügen Sie die Integrationskonstante hinzu:

$$\int{5 e^{5 s} \sin{\left(e^{5 s} \right)} d s} = - \cos{\left(e^{5 s} \right)}+C$$

$$$\int 5 e^{5 s} \sin{\left(e^{5 s} \right)}\, ds = - \cos{\left(e^{5 s} \right)} + C$$$A