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Beliebt Rechnen >

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

Lösung

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

Wende U-Substitution an

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

Multipliziere aus

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

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

\int 1 d u = u

\int \frac{2}{u} d u = 2 ln ∣ u ∣

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

= u + 2 ln ∣ u ∣ - \frac{1}{u}

Setze in

u = x - 1

ein

= 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

Beliebte Beispiele

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

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\int \frac{1}{x ^{2} - 25} d x

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\int \frac{x ^{2} - 64}{x} d x