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Beliebt Rechnen >

integral of (x^2+12x+12)/(x^3-4x)

Lösung

\int \frac{x ^{2} + 12 x + 12}{x ^{3} - 4 x} d x

Lösung

- 3 ln ∣ x ∣ - ln ∣ x + 2 ∣ + 5 ln ∣ x - 2 ∣ + C

Schritte zur Lösung

\int \frac{x ^{2} + 12 x + 12}{x ^{3} - 4 x} d x

Ermittle den Partialbruch von

\frac{x ^{2} + 12 x + 12}{x ^{3} - 4 x} : - \frac{3}{x} - \frac{1}{x + 2} + \frac{5}{x - 2}

= \int - \frac{3}{x} - \frac{1}{x + 2} + \frac{5}{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{3}{x} d x - \int \frac{1}{x + 2} d x + \int \frac{5}{x - 2} d x

\int \frac{3}{x} d x = 3 ln ∣ x ∣

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

\int \frac{5}{x - 2} d x = 5 ln ∣ x - 2 ∣

= - 3 ln ∣ x ∣ - ln ∣ x + 2 ∣ + 5 ln ∣ x - 2 ∣

Füge eine Konstante zur Lösung hinzu

= - 3 ln ∣ x ∣ - ln ∣ x + 2 ∣ + 5 ln ∣ x - 2 ∣ + C

Graph

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