TU Wien:Mathematik 2 UE (diverse)/Übungen SS07/Beispiel 37

Man berechne:

$\int \frac{dx}{2{\sin }^2(x){\cos }^2(x)}$

Lösungsvorschlag von leiwand

Partielle Integration

$\frac{1}{2}\int \frac{dx}{{\sin }^2(x){\cos }^2(x)}=$

$\frac{1}{2}(\frac{\tan (x)}{{\sin }^2(x)}+2\int \tan (x)\frac{\cos (x)}{{\sin }^3(x)}dx)=$

Tangens ausschreiben

$\frac{1}{2}\frac{\sin (x)}{\cos (x){\sin }^2(x)}+\int \frac{\sin (x)}{\cos (x)}\frac{\cos (x)}{{\sin }^3(x)}dx=$

Kürzen

$\frac{1}{2}\frac{1}{\cos (x)\sin (x)}+\int \frac{1}{{\sin }^2(x)}dx=$

$\frac{1}{2\cos (x)\sin (x)}-cot(x)=$

Summensatz

$\frac{1}{\sin (2x)}-cot(x)$