Integral von $$$e^{9} \sin{\left(7 x \right)} \cos{\left(7 x \right)}$$$

Bestimme $$$\int e^{9} \sin{\left(7 x \right)} \cos{\left(7 x \right)}\, dx$$$.

Wende die Konstantenfaktorregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ mit $$$c=e^{9}$$$ und $$$f{\left(x \right)} = \sin{\left(7 x \right)} \cos{\left(7 x \right)}$$$ an:

$${\color{red}{\int{e^{9} \sin{\left(7 x \right)} \cos{\left(7 x \right)} d x}}} = {\color{red}{e^{9} \int{\sin{\left(7 x \right)} \cos{\left(7 x \right)} d x}}}$$

Sei $$$u=\sin{\left(7 x \right)}$$$.

Dann $$$du=\left(\sin{\left(7 x \right)}\right)^{\prime }dx = 7 \cos{\left(7 x \right)} dx$$$ (die Schritte sind » zu sehen), und es gilt $$$\cos{\left(7 x \right)} dx = \frac{du}{7}$$$.

Also,

$$e^{9} {\color{red}{\int{\sin{\left(7 x \right)} \cos{\left(7 x \right)} d x}}} = e^{9} {\color{red}{\int{\frac{u}{7} d u}}}$$

Wende die Konstantenfaktorregel $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ mit $$$c=\frac{1}{7}$$$ und $$$f{\left(u \right)} = u$$$ an:

$$e^{9} {\color{red}{\int{\frac{u}{7} d u}}} = e^{9} {\color{red}{\left(\frac{\int{u d u}}{7}\right)}}$$

Wenden Sie die Potenzregel $$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ mit $$$n=1$$$ an:

$$\frac{e^{9} {\color{red}{\int{u d u}}}}{7}=\frac{e^{9} {\color{red}{\frac{u^{1 + 1}}{1 + 1}}}}{7}=\frac{e^{9} {\color{red}{\left(\frac{u^{2}}{2}\right)}}}{7}$$

Zur Erinnerung: $$$u=\sin{\left(7 x \right)}$$$:

$$\frac{e^{9} {\color{red}{u}}^{2}}{14} = \frac{e^{9} {\color{red}{\sin{\left(7 x \right)}}}^{2}}{14}$$

Daher,

$$\int{e^{9} \sin{\left(7 x \right)} \cos{\left(7 x \right)} d x} = \frac{e^{9} \sin^{2}{\left(7 x \right)}}{14}$$

Fügen Sie die Integrationskonstante hinzu:

$$\int{e^{9} \sin{\left(7 x \right)} \cos{\left(7 x \right)} d x} = \frac{e^{9} \sin^{2}{\left(7 x \right)}}{14}+C$$

$$$\int e^{9} \sin{\left(7 x \right)} \cos{\left(7 x \right)}\, dx = \frac{e^{9} \sin^{2}{\left(7 x \right)}}{14} + C$$$A