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Beliebt Rechnen >

integral of (x^2+x)/(x-1+x^3-x^2)

Lösung

\int \frac{x ^{2} + x}{x - 1 + x ^{3} - x ^{2}} d x

Lösung

ln ∣ x - 1 ∣ + arctan (x) + C

Schritte zur Lösung

\int \frac{x ^{2} + x}{x - 1 + x ^{3} - x ^{2}} d x

Ermittle den Partialbruch von

\frac{x ^{2} + x}{x - 1 + x ^{3} - x ^{2}} : \frac{1}{x - 1} + \frac{1}{x ^{2} + 1}

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

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

\int \frac{1}{x - 1} d x = ln ∣ x - 1 ∣

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

= ln ∣ x - 1 ∣ + arctan (x)

Füge eine Konstante zur Lösung hinzu

= ln ∣ x - 1 ∣ + arctan (x) + C

Graph

Beliebte Beispiele

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

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

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\int a^{2} - u^{2} d u

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\int e^{- 2 x} 2 x d x