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Beliebt Rechnen >

integral of 2/(x^3+1)

Lösung

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

Lösung

2 (\frac{1}{3} ln ∣ x + 1 ∣ + \frac{1}{6} (- ln \frac{4}{3} + \frac{4 x ^{2} - 4 x}{3} + 23 arctan (\frac{2 x - 1}{3}))) + C

Schritte zur Lösung

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

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

Ermittle den Partialbruch von

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

= 2 \cdot \int \frac{1}{3 ( x + 1 )} + \frac{- x + 2}{3 ( 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

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

\int \frac{1}{3 ( x + 1 )} d x = \frac{1}{3} ln ∣ x + 1 ∣

\int \frac{- x + 2}{3 ( x ^{2} - x + 1 )} d x = \frac{1}{6} (- ln \frac{4}{3} + \frac{4 x ^{2} - 4 x}{3} + 23 arctan (\frac{2 x - 1}{3}))

= 2 (\frac{1}{3} ln ∣ x + 1 ∣ + \frac{1}{6} (- ln \frac{4}{3} + \frac{4 x ^{2} - 4 x}{3} + 23 arctan (\frac{2 x - 1}{3})))

Füge eine Konstante zur Lösung hinzu

= 2 (\frac{1}{3} ln ∣ x + 1 ∣ + \frac{1}{6} (- ln \frac{4}{3} + \frac{4 x ^{2} - 4 x}{3} + 23 arctan (\frac{2 x - 1}{3}))) + C

Graph

Beliebte Beispiele

integral of e^{3x}x+e^x

\int e^{3 x} x + e^{x} d x

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

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

limit as x approaches 0+of-x+3

x \to 0 + lim (- x + 3)

(d^2y)/(dx^2)+25y=0

\frac{d ^{2} y}{d x ^{2}} + 25 y = 0

limit as x approaches 1-of (1-x)/(1+x)

x \to 1 - lim (\frac{1 - x}{1 + x})