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Beliebt Rechnen >

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

Lösung

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

Lösung

x + \frac{4}{5} ln ∣ x - 2 ∣ - \frac{9}{5} ln ∣ x + 3 ∣ + C

Schritte zur Lösung

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

Ermittle den Partialbruch von

\frac{x ^{2}}{x ^{2} + x - 6} : 1 + \frac{4}{5 ( x - 2 )} - \frac{9}{5 ( x + 3 )}

= \int 1 + \frac{4}{5 ( x - 2 )} - \frac{9}{5 ( 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 1 d x + \int \frac{4}{5 ( x - 2 )} d x - \int \frac{9}{5 ( x + 3 )} d x

\int 1 d x = x

\int \frac{4}{5 ( x - 2 )} d x = \frac{4}{5} ln ∣ x - 2 ∣

\int \frac{9}{5 ( x + 3 )} d x = \frac{9}{5} ln ∣ x + 3 ∣

= x + \frac{4}{5} ln ∣ x - 2 ∣ - \frac{9}{5} ln ∣ x + 3 ∣

Füge eine Konstante zur Lösung hinzu

= x + \frac{4}{5} ln ∣ x - 2 ∣ - \frac{9}{5} ln ∣ x + 3 ∣ + C

Graph

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