Integral von $$$\cos{\left(23 x \right)}$$$

Bestimme $$$\int \cos{\left(23 x \right)}\, dx$$$.

Sei $$$u=23 x$$$.

Dann $$$du=\left(23 x\right)^{\prime }dx = 23 dx$$$ (die Schritte sind » zu sehen), und es gilt $$$dx = \frac{du}{23}$$$.

Daher,

$${\color{red}{\int{\cos{\left(23 x \right)} d x}}} = {\color{red}{\int{\frac{\cos{\left(u \right)}}{23} d u}}}$$

Wende die Konstantenfaktorregel $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ mit $$$c=\frac{1}{23}$$$ und $$$f{\left(u \right)} = \cos{\left(u \right)}$$$ an:

$${\color{red}{\int{\frac{\cos{\left(u \right)}}{23} d u}}} = {\color{red}{\left(\frac{\int{\cos{\left(u \right)} d u}}{23}\right)}}$$

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

$$\frac{{\color{red}{\int{\cos{\left(u \right)} d u}}}}{23} = \frac{{\color{red}{\sin{\left(u \right)}}}}{23}$$

Zur Erinnerung: $$$u=23 x$$$:

$$\frac{\sin{\left({\color{red}{u}} \right)}}{23} = \frac{\sin{\left({\color{red}{\left(23 x\right)}} \right)}}{23}$$

Daher,

$$\int{\cos{\left(23 x \right)} d x} = \frac{\sin{\left(23 x \right)}}{23}$$

Fügen Sie die Integrationskonstante hinzu:

$$\int{\cos{\left(23 x \right)} d x} = \frac{\sin{\left(23 x \right)}}{23}+C$$

$$$\int \cos{\left(23 x \right)}\, dx = \frac{\sin{\left(23 x \right)}}{23} + C$$$A