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Beliebt Rechnen >

integral of (z^2)/((z-1)^3)

Lösung

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

Lösung

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

Schritte zur Lösung

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

Wende U-Substitution an

= \int \frac{( u + 1 ) ^{2}}{u ^{3}} d u

Multipliziere aus

\frac{( u + 1 ) ^{2}}{u ^{3}} : \frac{1}{u} + \frac{2}{u ^{2}} + \frac{1}{u ^{3}}

= \int \frac{1}{u} + \frac{2}{u ^{2}} + \frac{1}{u ^{3}} d u

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{1}{u} d u + \int \frac{2}{u ^{2}} d u + \int \frac{1}{u ^{3}} d u

\int \frac{1}{u} d u = ln ∣ u ∣

\int \frac{2}{u ^{2}} d u = - \frac{2}{u}

\int \frac{1}{u ^{3}} d u = - \frac{1}{2 u ^{2}}

= ln ∣ u ∣ - \frac{2}{u} - \frac{1}{2 u ^{2}}

Setze in

u = z - 1

ein

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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