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Beliebt Rechnen >

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

Lösung

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

Lösung

ln ∣ x - 2 ∣ + \frac{1}{2} ln 4 x^{2} + 4 x + 4 - 3 arctan (\frac{1}{3} (2 x + 1)) + C

Schritte zur Lösung

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

Ermittle den Partialbruch von

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

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

\int \frac{1}{x - 2} d x = ln ∣ x - 2 ∣

\int \frac{x - 1}{x ^{2} + x + 1} d x = \frac{1}{2} ln 4 x^{2} + 4 x + 4 - 3 arctan (\frac{1}{3} (2 x + 1))

= ln ∣ x - 2 ∣ + \frac{1}{2} ln 4 x^{2} + 4 x + 4 - 3 arctan (\frac{1}{3} (2 x + 1))

Füge eine Konstante zur Lösung hinzu

= ln ∣ x - 2 ∣ + \frac{1}{2} ln 4 x^{2} + 4 x + 4 - 3 arctan (\frac{1}{3} (2 x + 1)) + C

Graph

Beliebte Beispiele

integral of (2x+3)^9

\int (2 x + 3)^{9} d x

integral of x^2e^{-x/y}

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

integral of (4x+1)/(x^2+2x+3)

\int \frac{4 x + 1}{x ^{2} + 2 x + 3} d x

integral of (6x+6)/((x^2+2x)^2)

\int \frac{6 x + 6}{( x ^{2} + 2 x ) ^{2}} d x

integral of 10x+3

\int 10 x + 3 d x