integral of (3x+11)/((x^2-x-6))

Lösung

\int \frac{3 x + 11}{( x ^{2} - x - 6 )} d x

- ln ∣ x + 2 ∣ + 4 ln ∣ x - 3 ∣ + C

Schritte zur Lösung

\int \frac{3 x + 11}{x ^{2} - x - 6} d x

Ermittle den Partialbruch von

\frac{3 x + 11}{x ^{2} - x - 6} : - \frac{1}{x + 2} + \frac{4}{x - 3}

= \int - \frac{1}{x + 2} + \frac{4}{x - 3} 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{1}{x + 2} d x + \int \frac{4}{x - 3} d x

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

\int \frac{4}{x - 3} d x = 4 ln ∣ x - 3 ∣

= - ln ∣ x + 2 ∣ + 4 ln ∣ x - 3 ∣

Füge eine Konstante zur Lösung hinzu

= - ln ∣ x + 2 ∣ + 4 ln ∣ x - 3 ∣ + C

\int x - \frac{1}{3 x} d x

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