Integral von $$$\cos{\left(4 \theta \right)}$$$

Bestimme $$$\int \cos{\left(4 \theta \right)}\, d\theta$$$.

Sei $$$u=4 \theta$$$.

Dann $$$du=\left(4 \theta\right)^{\prime }d\theta = 4 d\theta$$$ (die Schritte sind » zu sehen), und es gilt $$$d\theta = \frac{du}{4}$$$.

Das Integral wird zu

$${\color{red}{\int{\cos{\left(4 \theta \right)} d \theta}}} = {\color{red}{\int{\frac{\cos{\left(u \right)}}{4} d u}}}$$

Wende die Konstantenfaktorregel $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ mit $$$c=\frac{1}{4}$$$ und $$$f{\left(u \right)} = \cos{\left(u \right)}$$$ an:

$${\color{red}{\int{\frac{\cos{\left(u \right)}}{4} d u}}} = {\color{red}{\left(\frac{\int{\cos{\left(u \right)} d u}}{4}\right)}}$$

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

$$\frac{{\color{red}{\int{\cos{\left(u \right)} d u}}}}{4} = \frac{{\color{red}{\sin{\left(u \right)}}}}{4}$$

Zur Erinnerung: $$$u=4 \theta$$$:

$$\frac{\sin{\left({\color{red}{u}} \right)}}{4} = \frac{\sin{\left({\color{red}{\left(4 \theta\right)}} \right)}}{4}$$

Daher,

$$\int{\cos{\left(4 \theta \right)} d \theta} = \frac{\sin{\left(4 \theta \right)}}{4}$$

Fügen Sie die Integrationskonstante hinzu:

$$\int{\cos{\left(4 \theta \right)} d \theta} = \frac{\sin{\left(4 \theta \right)}}{4}+C$$

$$$\int \cos{\left(4 \theta \right)}\, d\theta = \frac{\sin{\left(4 \theta \right)}}{4} + C$$$A