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Beliebt Rechnen >

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

Lösung

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

Lösung

\frac{1}{3} ln x^{2} + 2 + \frac{1}{3} ln ∣ x ∣ + C

Schritte zur Lösung

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

Multipliziere aus

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

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

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

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

Vereinfache

\frac{1}{2} ln x^{2} + 2 + \frac{1}{6} (2 ln ∣ x ∣ - ln x^{2} + 2) : \frac{1}{3} ln x^{2} + 2 + \frac{1}{3} ln ∣ x ∣

= \frac{1}{3} ln x^{2} + 2 + \frac{1}{3} ln ∣ x ∣

Füge eine Konstante zur Lösung hinzu

= \frac{1}{3} ln x^{2} + 2 + \frac{1}{3} ln ∣ x ∣ + C

Graph

Beliebte Beispiele

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