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

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

Schreibe den Integranden in Abhängigkeit von der Sekans um:

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

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

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

Daher,

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

Fügen Sie die Integrationskonstante hinzu:

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

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