Integral von $$$\frac{1}{4 \cos^{2}{\left(x \right)}}$$$

Bestimme $$$\int \frac{1}{4 \cos^{2}{\left(x \right)}}\, dx$$$.

Wende die Konstantenfaktorregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ mit $$$c=\frac{1}{4}$$$ und $$$f{\left(x \right)} = \frac{1}{\cos^{2}{\left(x \right)}}$$$ an:

$${\color{red}{\int{\frac{1}{4 \cos^{2}{\left(x \right)}} d x}}} = {\color{red}{\left(\frac{\int{\frac{1}{\cos^{2}{\left(x \right)}} d x}}{4}\right)}}$$

Schreibe den Integranden in Abhängigkeit von der Sekans um:

$$\frac{{\color{red}{\int{\frac{1}{\cos^{2}{\left(x \right)}} d x}}}}{4} = \frac{{\color{red}{\int{\sec^{2}{\left(x \right)} d x}}}}{4}$$

Das Integral von $$$\sec^{2}{\left(x \right)}$$$ ist $$$\int{\sec^{2}{\left(x \right)} d x} = \tan{\left(x \right)}$$$:

$$\frac{{\color{red}{\int{\sec^{2}{\left(x \right)} d x}}}}{4} = \frac{{\color{red}{\tan{\left(x \right)}}}}{4}$$

Daher,

$$\int{\frac{1}{4 \cos^{2}{\left(x \right)}} d x} = \frac{\tan{\left(x \right)}}{4}$$

Fügen Sie die Integrationskonstante hinzu:

$$\int{\frac{1}{4 \cos^{2}{\left(x \right)}} d x} = \frac{\tan{\left(x \right)}}{4}+C$$

$$$\int \frac{1}{4 \cos^{2}{\left(x \right)}}\, dx = \frac{\tan{\left(x \right)}}{4} + C$$$A