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Beliebt Rechnen >

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

Lösung

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

Lösung

2 ln ∣ x - 1 ∣ - \frac{4}{x - 1} + C

Schritte zur Lösung

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

Vervollständige das Quadrat

x^{2} - 2 x + 1 : (x - 1)^{2}

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

Wende U-Substitution an

= \int \frac{2 u + 4}{u ^{2}} d u

Multipliziere aus

\frac{2 u + 4}{u ^{2}} : \frac{2}{u} + \frac{4}{u ^{2}}

= \int \frac{2}{u} + \frac{4}{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 \frac{2}{u} d u + \int \frac{4}{u ^{2}} d u

\int \frac{2}{u} d u = 2 ln ∣ u ∣

\int \frac{4}{u ^{2}} d u = - \frac{4}{u}

= 2 ln ∣ u ∣ - \frac{4}{u}

Setze in

u = x - 1

ein

= 2 ln ∣ x - 1 ∣ - \frac{4}{x - 1}

Füge eine Konstante zur Lösung hinzu

= 2 ln ∣ x - 1 ∣ - \frac{4}{x - 1} + C

Graph

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