Integral von $$$5 \sin{\left(5 x \right)}$$$

Bestimme $$$\int 5 \sin{\left(5 x \right)}\, dx$$$.

Wende die Konstantenfaktorregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ mit $$$c=5$$$ und $$$f{\left(x \right)} = \sin{\left(5 x \right)}$$$ an:

$${\color{red}{\int{5 \sin{\left(5 x \right)} d x}}} = {\color{red}{\left(5 \int{\sin{\left(5 x \right)} d x}\right)}}$$

Sei $$$u=5 x$$$.

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

Daher,

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

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

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

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

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

Zur Erinnerung: $$$u=5 x$$$:

$$- \cos{\left({\color{red}{u}} \right)} = - \cos{\left({\color{red}{\left(5 x\right)}} \right)}$$

Daher,

$$\int{5 \sin{\left(5 x \right)} d x} = - \cos{\left(5 x \right)}$$

Fügen Sie die Integrationskonstante hinzu:

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

$$$\int 5 \sin{\left(5 x \right)}\, dx = - \cos{\left(5 x \right)} + C$$$A