Integral von $$$\cot^{2}{\left(\theta \right)}$$$

Bestimme $$$\int \cot^{2}{\left(\theta \right)}\, d\theta$$$.

Sei $$$u=\cot{\left(\theta \right)}$$$.

Dann $$$du=\left(\cot{\left(\theta \right)}\right)^{\prime }d\theta = - \csc^{2}{\left(\theta \right)} d\theta$$$ (die Schritte sind » zu sehen), und es gilt $$$\csc^{2}{\left(\theta \right)} d\theta = - du$$$.

Also,

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

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

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

Forme den Bruch um und zerlege ihn:

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

Gliedweise integrieren:

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

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

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

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

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

Zur Erinnerung: $$$u=\cot{\left(\theta \right)}$$$:

$$\operatorname{atan}{\left({\color{red}{u}} \right)} - {\color{red}{u}} = \operatorname{atan}{\left({\color{red}{\cot{\left(\theta \right)}}} \right)} - {\color{red}{\cot{\left(\theta \right)}}}$$

Daher,

$$\int{\cot^{2}{\left(\theta \right)} d \theta} = - \cot{\left(\theta \right)} + \operatorname{atan}{\left(\cot{\left(\theta \right)} \right)}$$

Fügen Sie die Integrationskonstante hinzu:

$$\int{\cot^{2}{\left(\theta \right)} d \theta} = - \cot{\left(\theta \right)} + \operatorname{atan}{\left(\cot{\left(\theta \right)} \right)}+C$$

$$$\int \cot^{2}{\left(\theta \right)}\, d\theta = \left(- \cot{\left(\theta \right)} + \operatorname{atan}{\left(\cot{\left(\theta \right)} \right)}\right) + C$$$A