Integral von $$$\frac{\tan{\left(x \right)}}{\cos{\left(x \right)}}$$$

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

Schreiben Sie den Integranden um:

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

Sei $$$u=\sec{\left(x \right)}$$$.

Dann $$$du=\left(\sec{\left(x \right)}\right)^{\prime }dx = \tan{\left(x \right)} \sec{\left(x \right)} dx$$$ (die Schritte sind » zu sehen), und es gilt $$$\tan{\left(x \right)} \sec{\left(x \right)} dx = du$$$.

Also,

$${\color{red}{\int{\tan{\left(x \right)} \sec{\left(x \right)} d x}}} = {\color{red}{\int{1 d u}}}$$

Wenden Sie die Konstantenregel $$$\int c\, du = c u$$$ mit $$$c=1$$$ an:

$${\color{red}{\int{1 d u}}} = {\color{red}{u}}$$

Zur Erinnerung: $$$u=\sec{\left(x \right)}$$$:

$${\color{red}{u}} = {\color{red}{\sec{\left(x \right)}}}$$

Daher,

$$\int{\frac{\tan{\left(x \right)}}{\cos{\left(x \right)}} d x} = \sec{\left(x \right)}$$

Fügen Sie die Integrationskonstante hinzu:

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

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