integral of (1-1/z)^2 1/(z^2)

Lösung

\int (1 - \frac{1}{z})^{2} \frac{1}{z ^{2}} d z

- \frac{( z - 1 ) ^{2}}{z ^{3}} - \frac{1}{z ^{2}} + \frac{2}{3 z ^{3}} + C

Schritte zur Lösung

\int (1 - \frac{1}{z})^{2} \frac{1}{z ^{2}} d z

Wende die partielle Integration an

= - \frac{( z - 1 ) ^{2}}{z ^{3}} - \int - \frac{2 ( z - 1 )}{z ^{4}} d z

\int - \frac{2 ( z - 1 )}{z ^{4}} d z = \frac{1}{z ^{2}} - \frac{2}{3 z ^{3}}

= - \frac{( z - 1 ) ^{2}}{z ^{3}} - (\frac{1}{z ^{2}} - \frac{2}{3 z ^{3}})

Vereinfache

= - \frac{( z - 1 ) ^{2}}{z ^{3}} - \frac{1}{z ^{2}} + \frac{2}{3 z ^{3}}

Füge eine Konstante zur Lösung hinzu

= - \frac{( z - 1 ) ^{2}}{z ^{3}} - \frac{1}{z ^{2}} + \frac{2}{3 z ^{3}} + C

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