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Beliebt Rechnen >

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

Lösung

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

Lösung

x - 1 + 2 ln ∣ x - 1 ∣ + \frac{1}{x - 1} + C

Schritte zur Lösung

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

Multipliziere aus

\frac{x ^{2} - 2}{( x - 1 ) ^{2}} : \frac{x ^{2}}{( x - 1 ) ^{2}} - \frac{2}{( x - 1 ) ^{2}}

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

\int \frac{x ^{2}}{( x - 1 ) ^{2}} d x = x - 1 + 2 ln ∣ x - 1 ∣ - \frac{1}{x - 1}

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

= x - 1 + 2 ln ∣ x - 1 ∣ - \frac{1}{x - 1} - (- \frac{2}{x - 1})

Vereinfache

= x - 1 + 2 ln ∣ x - 1 ∣ + \frac{1}{x - 1}

Füge eine Konstante zur Lösung hinzu

= x - 1 + 2 ln ∣ x - 1 ∣ + \frac{1}{x - 1} + C

Graph

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