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Beliebt Rechnen >

integral of (2-v)/(v^2+2v-3)

Lösung

\int \frac{2 - v}{v ^{2} + 2 v - 3} d v

Lösung

\frac{1}{4} ln ∣ v - 1 ∣ - \frac{5}{4} ln ∣ v + 3 ∣ + C

Schritte zur Lösung

\int \frac{2 - v}{v ^{2} + 2 v - 3} d v

Ermittle den Partialbruch von

\frac{2 - v}{v ^{2} + 2 v - 3} : \frac{1}{4 ( v - 1 )} - \frac{5}{4 ( v + 3 )}

= \int \frac{1}{4 ( v - 1 )} - \frac{5}{4 ( v + 3 )} d v

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}{4 ( v - 1 )} d v - \int \frac{5}{4 ( v + 3 )} d v

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

\int \frac{5}{4 ( v + 3 )} d v = \frac{5}{4} ln ∣ v + 3 ∣

= \frac{1}{4} ln ∣ v - 1 ∣ - \frac{5}{4} ln ∣ v + 3 ∣

Füge eine Konstante zur Lösung hinzu

= \frac{1}{4} ln ∣ v - 1 ∣ - \frac{5}{4} ln ∣ v + 3 ∣ + C

Graph

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