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Beliebt Rechnen >

integral of (2r+1)/(r^2-3r+2)

Lösung

\int \frac{2 r + 1}{r ^{2} - 3 r + 2} d r

Lösung

- 3 ln ∣ r - 1 ∣ + 5 ln ∣ r - 2 ∣ + C

Schritte zur Lösung

\int \frac{2 r + 1}{r ^{2} - 3 r + 2} d r

Ermittle den Partialbruch von

\frac{2 r + 1}{r ^{2} - 3 r + 2} : - \frac{3}{r - 1} + \frac{5}{r - 2}

= \int - \frac{3}{r - 1} + \frac{5}{r - 2} d r

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}{r - 1} d r + \int \frac{5}{r - 2} d r

\int \frac{3}{r - 1} d r = 3 ln ∣ r - 1 ∣

\int \frac{5}{r - 2} d r = 5 ln ∣ r - 2 ∣

= - 3 ln ∣ r - 1 ∣ + 5 ln ∣ r - 2 ∣

Füge eine Konstante zur Lösung hinzu

= - 3 ln ∣ r - 1 ∣ + 5 ln ∣ r - 2 ∣ + C

Graph

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