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Beliebt Rechnen >

integral of (-2(x-9))/((x+5)(x-2))

Lösung

\int \frac{- 2 ( x - 9 )}{( x + 5 ) ( x - 2 )} d x

Lösung

- 2 (2 ln ∣ x + 5 ∣ - ln ∣ x - 2 ∣) + C

Schritte zur Lösung

\int \frac{- 2 ( x - 9 )}{( x + 5 ) ( x - 2 )} d x

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - 2 \cdot \int \frac{x - 9}{( x + 5 ) ( x - 2 )} d x

Ermittle den Partialbruch von

\frac{x - 9}{( x + 5 ) ( x - 2 )} : \frac{2}{x + 5} - \frac{1}{x - 2}

= - 2 \cdot \int \frac{2}{x + 5} - \frac{1}{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

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

\int \frac{2}{x + 5} d x = 2 ln ∣ x + 5 ∣

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

= - 2 (2 ln ∣ x + 5 ∣ - ln ∣ x - 2 ∣)

Füge eine Konstante zur Lösung hinzu

= - 2 (2 ln ∣ x + 5 ∣ - ln ∣ x - 2 ∣) + C

Graph

Beliebte Beispiele

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