Integral von $$$\tan{\left(x \right)} \sec^{p}{\left(x \right)}$$$ nach $$$x$$$

Bestimme $$$\int \tan{\left(x \right)} \sec^{p}{\left(x \right)}\, dx$$$.

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 $$$\tan{\left(x \right)} \sec{\left(x \right)} dx = du$$$.

Das Integral lässt sich umschreiben als

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

Wenden Sie die Potenzregel $$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ mit $$$n=p - 1$$$ an:

$${\color{red}{\int{u^{p - 1} d u}}}={\color{red}{\frac{u^{\left(p - 1\right) + 1}}{\left(p - 1\right) + 1}}}={\color{red}{\frac{u^{p}}{p}}}$$

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

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

Daher,

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

Fügen Sie die Integrationskonstante hinzu:

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

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