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

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