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Beliebt Rechnen >

integral of (x^2+2x-1)/(x^3-27)

Lösung

\int \frac{x ^{2} + 2 x - 1}{x ^{3} - 27} d x

Lösung

\frac{1}{3} ln x^{3} - 27 + \frac{5}{27} ln ∣ x - 3 ∣ + \frac{1}{54} ln 4 x^{2} + 12 x + 36 + \frac{1}{9 3} arctan (\frac{2}{3 3} x + \frac{1}{3}) + \frac{1}{9} (- ln 4 x^{2} + 12 x + 36 + 23 arctan (\frac{2}{3 3} x + \frac{1}{3})) + C

Schritte zur Lösung

\int \frac{x ^{2} + 2 x - 1}{x ^{3} - 27} d x

Multipliziere aus

\frac{x ^{2} + 2 x - 1}{x ^{3} - 27} : \frac{x ^{2}}{x ^{3} - 27} + \frac{2 x}{x ^{3} - 27} - \frac{1}{x ^{3} - 27}

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{x ^{2}}{x ^{3} - 27} d x + \int \frac{2 x}{x ^{3} - 27} d x - \int \frac{1}{x ^{3} - 27} d x

\int \frac{x ^{2}}{x ^{3} - 27} d x = \frac{1}{3} ln x^{3} - 27

\int \frac{2 x}{x ^{3} - 27} d x = 2 (\frac{1}{9} ln ∣ x - 3 ∣ + \frac{1}{18} (- ln 4 x^{2} + 12 x + 36 + 23 arctan (\frac{1}{3 3} (2 x + 3))))

\int \frac{1}{x ^{3} - 27} d x = \frac{1}{27} ln ∣ x - 3 ∣ - \frac{1}{54} ln 4 x^{2} + 12 x + 36 - \frac{1}{9 3} arctan (\frac{2}{3 3} x + \frac{1}{3})

= \frac{1}{3} ln x^{3} - 27 + 2 (\frac{1}{9} ln ∣ x - 3 ∣ + \frac{1}{18} (- ln 4 x^{2} + 12 x + 36 + 23 arctan (\frac{1}{3 3} (2 x + 3)))) - (\frac{1}{27} ln ∣ x - 3 ∣ - \frac{1}{54} ln 4 x^{2} + 12 x + 36 - \frac{1}{9 3} arctan (\frac{2}{3 3} x + \frac{1}{3}))

Vereinfache

\frac{1}{3} ln x^{3} - 27 + 2 (\frac{1}{9} ln ∣ x - 3 ∣ + \frac{1}{18} (- ln 4 x^{2} + 12 x + 36 + 23 arctan (\frac{1}{3 3} (2 x + 3)))) - (\frac{1}{27} ln ∣ x - 3 ∣ - \frac{1}{54} ln 4 x^{2} + 12 x + 36 - \frac{1}{9 3} arctan (\frac{2}{3 3} x + \frac{1}{3})) : \frac{1}{3} ln x^{3} - 27 + \frac{5}{27} ln ∣ x - 3 ∣ + \frac{1}{54} ln 4 x^{2} + 12 x + 36 + \frac{1}{9 3} arctan (\frac{2}{3 3} x + \frac{1}{3}) + \frac{1}{9} (- ln 4 x^{2} + 12 x + 36 + 23 arctan (\frac{2}{3 3} x + \frac{1}{3}))

= \frac{1}{3} ln x^{3} - 27 + \frac{5}{27} ln ∣ x - 3 ∣ + \frac{1}{54} ln 4 x^{2} + 12 x + 36 + \frac{1}{9 3} arctan (\frac{2}{3 3} x + \frac{1}{3}) + \frac{1}{9} (- ln 4 x^{2} + 12 x + 36 + 23 arctan (\frac{2}{3 3} x + \frac{1}{3}))

Füge eine Konstante zur Lösung hinzu

= \frac{1}{3} ln x^{3} - 27 + \frac{5}{27} ln ∣ x - 3 ∣ + \frac{1}{54} ln 4 x^{2} + 12 x + 36 + \frac{1}{9 3} arctan (\frac{2}{3 3} x + \frac{1}{3}) + \frac{1}{9} (- ln 4 x^{2} + 12 x + 36 + 23 arctan (\frac{2}{3 3} x + \frac{1}{3})) + C

Graph

Beliebte Beispiele

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

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

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