integral of (-3x+7)/(x^2+x-2)

Lösung

\int \frac{- 3 x + 7}{x ^{2} + x - 2} d x

\frac{4}{3} ln ∣ x - 1 ∣ - \frac{13}{3} ln ∣ x + 2 ∣ + C

Schritte zur Lösung

\int \frac{- 3 x + 7}{x ^{2} + x - 2} d x

Ermittle den Partialbruch von

\frac{- 3 x + 7}{x ^{2} + x - 2} : \frac{4}{3 ( x - 1 )} - \frac{13}{3 ( x + 2 )}

= \int \frac{4}{3 ( x - 1 )} - \frac{13}{3 ( 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{4}{3 ( x - 1 )} d x - \int \frac{13}{3 ( x + 2 )} d x

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

\int \frac{13}{3 ( x + 2 )} d x = \frac{13}{3} ln ∣ x + 2 ∣

= \frac{4}{3} ln ∣ x - 1 ∣ - \frac{13}{3} ln ∣ x + 2 ∣

Füge eine Konstante zur Lösung hinzu

= \frac{4}{3} ln ∣ x - 1 ∣ - \frac{13}{3} ln ∣ x + 2 ∣ + C

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