Integral von $$$\tan^{2}{\left(2 x \right)}$$$

Bestimme $$$\int \tan^{2}{\left(2 x \right)}\, dx$$$.

Sei $$$u=2 x$$$.

Dann $$$du=\left(2 x\right)^{\prime }dx = 2 dx$$$ (die Schritte sind » zu sehen), und es gilt $$$dx = \frac{du}{2}$$$.

Also,

$${\color{red}{\int{\tan^{2}{\left(2 x \right)} d x}}} = {\color{red}{\int{\frac{\tan^{2}{\left(u \right)}}{2} d u}}}$$

Wende die Konstantenfaktorregel $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ mit $$$c=\frac{1}{2}$$$ und $$$f{\left(u \right)} = \tan^{2}{\left(u \right)}$$$ an:

$${\color{red}{\int{\frac{\tan^{2}{\left(u \right)}}{2} d u}}} = {\color{red}{\left(\frac{\int{\tan^{2}{\left(u \right)} d u}}{2}\right)}}$$

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

Dann gelten $$$u=\operatorname{atan}{\left(v \right)}$$$ und $$$du=\left(\operatorname{atan}{\left(v \right)}\right)^{\prime }dv = \frac{dv}{v^{2} + 1}$$$ (die Schritte sind » zu sehen).

Das Integral lässt sich umschreiben als

$$\frac{{\color{red}{\int{\tan^{2}{\left(u \right)} d u}}}}{2} = \frac{{\color{red}{\int{\frac{v^{2}}{v^{2} + 1} d v}}}}{2}$$

Forme den Bruch um und zerlege ihn:

$$\frac{{\color{red}{\int{\frac{v^{2}}{v^{2} + 1} d v}}}}{2} = \frac{{\color{red}{\int{\left(1 - \frac{1}{v^{2} + 1}\right)d v}}}}{2}$$

Gliedweise integrieren:

$$\frac{{\color{red}{\int{\left(1 - \frac{1}{v^{2} + 1}\right)d v}}}}{2} = \frac{{\color{red}{\left(\int{1 d v} - \int{\frac{1}{v^{2} + 1} d v}\right)}}}{2}$$

Wenden Sie die Konstantenregel $$$\int c\, dv = c v$$$ mit $$$c=1$$$ an:

$$- \frac{\int{\frac{1}{v^{2} + 1} d v}}{2} + \frac{{\color{red}{\int{1 d v}}}}{2} = - \frac{\int{\frac{1}{v^{2} + 1} d v}}{2} + \frac{{\color{red}{v}}}{2}$$

Das Integral von $$$\frac{1}{v^{2} + 1}$$$ ist $$$\int{\frac{1}{v^{2} + 1} d v} = \operatorname{atan}{\left(v \right)}$$$:

$$\frac{v}{2} - \frac{{\color{red}{\int{\frac{1}{v^{2} + 1} d v}}}}{2} = \frac{v}{2} - \frac{{\color{red}{\operatorname{atan}{\left(v \right)}}}}{2}$$

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

$$- \frac{\operatorname{atan}{\left({\color{red}{v}} \right)}}{2} + \frac{{\color{red}{v}}}{2} = - \frac{\operatorname{atan}{\left({\color{red}{\tan{\left(u \right)}}} \right)}}{2} + \frac{{\color{red}{\tan{\left(u \right)}}}}{2}$$

Zur Erinnerung: $$$u=2 x$$$:

$$\frac{\tan{\left({\color{red}{u}} \right)}}{2} - \frac{\operatorname{atan}{\left(\tan{\left({\color{red}{u}} \right)} \right)}}{2} = \frac{\tan{\left({\color{red}{\left(2 x\right)}} \right)}}{2} - \frac{\operatorname{atan}{\left(\tan{\left({\color{red}{\left(2 x\right)}} \right)} \right)}}{2}$$

Daher,

$$\int{\tan^{2}{\left(2 x \right)} d x} = \frac{\tan{\left(2 x \right)}}{2} - \frac{\operatorname{atan}{\left(\tan{\left(2 x \right)} \right)}}{2}$$

Vereinfachen:

$$\int{\tan^{2}{\left(2 x \right)} d x} = - x + \frac{\tan{\left(2 x \right)}}{2}$$

Fügen Sie die Integrationskonstante hinzu:

$$\int{\tan^{2}{\left(2 x \right)} d x} = - x + \frac{\tan{\left(2 x \right)}}{2}+C$$

$$$\int \tan^{2}{\left(2 x \right)}\, dx = \left(- x + \frac{\tan{\left(2 x \right)}}{2}\right) + C$$$A