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Beliebt Rechnen >

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

Lösung

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

Lösung

- 2 ln ∣ x ∣ + \frac{2}{x} + 2 ln ∣ x - 2 ∣ + C

Schritte zur Lösung

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

Ermittle den Partialbruch von

\frac{2 x + 4}{x ^{3} - 2 x ^{2}} : - \frac{2}{x} - \frac{2}{x ^{2}} + \frac{2}{x - 2}

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

\int \frac{2}{x} d x = 2 ln ∣ x ∣

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

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

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

Vereinfache

= - 2 ln ∣ x ∣ + \frac{2}{x} + 2 ln ∣ x - 2 ∣

Füge eine Konstante zur Lösung hinzu

= - 2 ln ∣ x ∣ + \frac{2}{x} + 2 ln ∣ x - 2 ∣ + C

Graph

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