Integral von $$$\frac{\sec{\left(x \right)}}{\cos{\left(x \right)}}$$$
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Ihre Eingabe
Bestimme $$$\int \frac{\sec{\left(x \right)}}{\cos{\left(x \right)}}\, dx$$$.
Lösung
Schreiben Sie den Integranden um:
$${\color{red}{\int{\frac{\sec{\left(x \right)}}{\cos{\left(x \right)}} d x}}} = {\color{red}{\int{\sec^{2}{\left(x \right)} d x}}}$$
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,
$$\int{\frac{\sec{\left(x \right)}}{\cos{\left(x \right)}} d x} = \tan{\left(x \right)}$$
Fügen Sie die Integrationskonstante hinzu:
$$\int{\frac{\sec{\left(x \right)}}{\cos{\left(x \right)}} d x} = \tan{\left(x \right)}+C$$
Antwort
$$$\int \frac{\sec{\left(x \right)}}{\cos{\left(x \right)}}\, dx = \tan{\left(x \right)} + C$$$A