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Beliebt Rechnen >

integral of ((2x))/((x-2)(x+1))

Lösung

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

Lösung

2 (\frac{2}{3} ln ∣ x - 2 ∣ + \frac{1}{3} ln ∣ x + 1 ∣) + C

Schritte zur Lösung

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

Entferne die Konstante:

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

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

Ermittle den Partialbruch von

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

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

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

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

= 2 (\frac{2}{3} ln ∣ x - 2 ∣ + \frac{1}{3} ln ∣ x + 1 ∣)

Füge eine Konstante zur Lösung hinzu

= 2 (\frac{2}{3} ln ∣ x - 2 ∣ + \frac{1}{3} ln ∣ x + 1 ∣) + C

Graph

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