Question 14028. (Q1509575)

Behandlung des Integrals: \[ \int_0^{\pi} \frac{\cos (p/q) \varphi \cdot d\varphi}{1+ 2t \cos \varphi + t^2}, \] wenn \(t<1\), \(p\) und \(q\) ganze Zahlen, \(p<q\). Anwendung: \[ \int_0^{\pi} \frac{\cos \frac 12 \varphi d\varphi}{1+ 2t \cos \varphi + t^2} = \frac{2}{1+t}\;\frac{\text{artgh} \sqrt t}{\sqrt t}\,, \] \[ \int_0^{\pi} \text{arctg} \left( \frac{2t \sin \varphi}{1-t^2} \right) \frac{d\varphi}{\sin \frac 12 \varphi} = 8\,\text{arctg}\, \sqrt t \cdot \text{artgh} \,\sqrt t . \]