Integral von $$$2 \cos{\left(2 x \right)}$$$

Bestimme $$$\int 2 \cos{\left(2 x \right)}\, dx$$$.

Wende die Konstantenfaktorregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ mit $$$c=2$$$ und $$$f{\left(x \right)} = \cos{\left(2 x \right)}$$$ an:

$${\color{red}{\int{2 \cos{\left(2 x \right)} d x}}} = {\color{red}{\left(2 \int{\cos{\left(2 x \right)} d x}\right)}}$$

Sei $$$u=2 x$$$.

Dann $$$du=\left(2 x\right)^{\prime }dx = 2 dx$$$ (die Schritte sind » zu sehen), und es gilt $$$dx = \frac{du}{2}$$$.

Also,

$$2 {\color{red}{\int{\cos{\left(2 x \right)} d x}}} = 2 {\color{red}{\int{\frac{\cos{\left(u \right)}}{2} d u}}}$$

Wende die Konstantenfaktorregel $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ mit $$$c=\frac{1}{2}$$$ und $$$f{\left(u \right)} = \cos{\left(u \right)}$$$ an:

$$2 {\color{red}{\int{\frac{\cos{\left(u \right)}}{2} d u}}} = 2 {\color{red}{\left(\frac{\int{\cos{\left(u \right)} d u}}{2}\right)}}$$

Das Integral des Kosinus ist $$$\int{\cos{\left(u \right)} d u} = \sin{\left(u \right)}$$$:

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

Zur Erinnerung: $$$u=2 x$$$:

$$\sin{\left({\color{red}{u}} \right)} = \sin{\left({\color{red}{\left(2 x\right)}} \right)}$$

Daher,

$$\int{2 \cos{\left(2 x \right)} d x} = \sin{\left(2 x \right)}$$

Fügen Sie die Integrationskonstante hinzu:

$$\int{2 \cos{\left(2 x \right)} d x} = \sin{\left(2 x \right)}+C$$

$$$\int 2 \cos{\left(2 x \right)}\, dx = \sin{\left(2 x \right)} + C$$$A