Integral von $$$\tan{\left(7 x \right)} \sec^{5}{\left(7 x \right)}$$$

Bestimme $$$\int \tan{\left(7 x \right)} \sec^{5}{\left(7 x \right)}\, dx$$$.

Sei $$$u=7 x$$$.

Dann $$$du=\left(7 x\right)^{\prime }dx = 7 dx$$$ (die Schritte sind » zu sehen), und es gilt $$$dx = \frac{du}{7}$$$.

Also,

$${\color{red}{\int{\tan{\left(7 x \right)} \sec^{5}{\left(7 x \right)} d x}}} = {\color{red}{\int{\frac{\tan{\left(u \right)} \sec^{5}{\left(u \right)}}{7} d u}}}$$

Wende die Konstantenfaktorregel $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ mit $$$c=\frac{1}{7}$$$ und $$$f{\left(u \right)} = \tan{\left(u \right)} \sec^{5}{\left(u \right)}$$$ an:

$${\color{red}{\int{\frac{\tan{\left(u \right)} \sec^{5}{\left(u \right)}}{7} d u}}} = {\color{red}{\left(\frac{\int{\tan{\left(u \right)} \sec^{5}{\left(u \right)} d u}}{7}\right)}}$$

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

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

Das Integral wird zu

$$\frac{{\color{red}{\int{\tan{\left(u \right)} \sec^{5}{\left(u \right)} d u}}}}{7} = \frac{{\color{red}{\int{v^{4} d v}}}}{7}$$

Wenden Sie die Potenzregel $$$\int v^{n}\, dv = \frac{v^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ mit $$$n=4$$$ an:

$$\frac{{\color{red}{\int{v^{4} d v}}}}{7}=\frac{{\color{red}{\frac{v^{1 + 4}}{1 + 4}}}}{7}=\frac{{\color{red}{\left(\frac{v^{5}}{5}\right)}}}{7}$$

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

$$\frac{{\color{red}{v}}^{5}}{35} = \frac{{\color{red}{\sec{\left(u \right)}}}^{5}}{35}$$

Zur Erinnerung: $$$u=7 x$$$:

$$\frac{\sec^{5}{\left({\color{red}{u}} \right)}}{35} = \frac{\sec^{5}{\left({\color{red}{\left(7 x\right)}} \right)}}{35}$$

Daher,

$$\int{\tan{\left(7 x \right)} \sec^{5}{\left(7 x \right)} d x} = \frac{\sec^{5}{\left(7 x \right)}}{35}$$

Fügen Sie die Integrationskonstante hinzu:

$$\int{\tan{\left(7 x \right)} \sec^{5}{\left(7 x \right)} d x} = \frac{\sec^{5}{\left(7 x \right)}}{35}+C$$

$$$\int \tan{\left(7 x \right)} \sec^{5}{\left(7 x \right)}\, dx = \frac{\sec^{5}{\left(7 x \right)}}{35} + C$$$A