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Integral von $$$\frac{1}{\cos^{2}{\left(\theta \right)}}$$$
Verwandter Rechner: Rechner für bestimmte und uneigentliche Integrale
Ihre Eingabe
Bestimme $$$\int \frac{1}{\cos^{2}{\left(\theta \right)}}\, d\theta$$$.
Lösung
Schreibe den Integranden in Abhängigkeit von der Sekans um:
$${\color{red}{\int{\frac{1}{\cos^{2}{\left(\theta \right)}} d \theta}}} = {\color{red}{\int{\sec^{2}{\left(\theta \right)} d \theta}}}$$
Das Integral von $$$\sec^{2}{\left(\theta \right)}$$$ ist $$$\int{\sec^{2}{\left(\theta \right)} d \theta} = \tan{\left(\theta \right)}$$$:
$${\color{red}{\int{\sec^{2}{\left(\theta \right)} d \theta}}} = {\color{red}{\tan{\left(\theta \right)}}}$$
Daher,
$$\int{\frac{1}{\cos^{2}{\left(\theta \right)}} d \theta} = \tan{\left(\theta \right)}$$
Fügen Sie die Integrationskonstante hinzu:
$$\int{\frac{1}{\cos^{2}{\left(\theta \right)}} d \theta} = \tan{\left(\theta \right)}+C$$
Antwort
$$$\int \frac{1}{\cos^{2}{\left(\theta \right)}}\, d\theta = \tan{\left(\theta \right)} + C$$$A