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Beliebt Rechnen >

integral of (x+8)/(3x^2-12)

Lösung

\int \frac{x + 8}{3 x ^{2} - 12} d x

Lösung

- \frac{1}{2} ln ∣ x + 2 ∣ + \frac{5}{6} ln ∣ x - 2 ∣ + C

Schritte zur Lösung

\int \frac{x + 8}{3 x ^{2} - 12} d x

Ermittle den Partialbruch von

\frac{x + 8}{3 x ^{2} - 12} : - \frac{1}{2 ( x + 2 )} + \frac{5}{6 ( x - 2 )}

= \int - \frac{1}{2 ( x + 2 )} + \frac{5}{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{1}{2 ( x + 2 )} d x + \int \frac{5}{6 ( x - 2 )} d x

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

\int \frac{5}{6 ( x - 2 )} d x = \frac{5}{6} ln ∣ x - 2 ∣

= - \frac{1}{2} ln ∣ x + 2 ∣ + \frac{5}{6} ln ∣ x - 2 ∣

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{2} ln ∣ x + 2 ∣ + \frac{5}{6} ln ∣ x - 2 ∣ + C

Graph

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