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Beliebt Rechnen >

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

Lösung

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

Lösung

- \frac{3}{2} ln ∣ x ∣ + \frac{4}{3} ln ∣ x - 1 ∣ + \frac{1}{6} ln ∣ x + 2 ∣ + C

Schritte zur Lösung

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

Ermittle den Partialbruch von

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

= \int - \frac{3}{2 x} + \frac{4}{3 ( x - 1 )} + \frac{1}{6 ( 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}{2 x} d x + \int \frac{4}{3 ( x - 1 )} d x + \int \frac{1}{6 ( x + 2 )} d x

\int \frac{3}{2 x} d x = \frac{3}{2} ln ∣ x ∣

\int \frac{4}{3 ( x - 1 )} d x = \frac{4}{3} ln ∣ x - 1 ∣

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

= - \frac{3}{2} ln ∣ x ∣ + \frac{4}{3} ln ∣ x - 1 ∣ + \frac{1}{6} ln ∣ x + 2 ∣

Füge eine Konstante zur Lösung hinzu

= - \frac{3}{2} ln ∣ x ∣ + \frac{4}{3} ln ∣ x - 1 ∣ + \frac{1}{6} ln ∣ x + 2 ∣ + C

Graph

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