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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Wende U-Substitution an

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

Multipliziere aus

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

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

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

\int 2 d u = 2 u

\int \frac{2}{u} d u = 2 ln ∣ u ∣

= \frac{u ^{2}}{2} + 2 u + 2 ln ∣ u ∣

Setze in

u = x - 1

ein

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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