integral of v/(1-kv)

Lösung

\int \frac{v}{1 - k v} d v

\frac{1}{k ^{2}} (1 - k v - ln ∣ 1 - k v ∣) + C

Schritte zur Lösung

\int \frac{v}{1 - k v} d v

Wende U-Substitution an

= \int \frac{u - 1}{k ^{2} u} d u

Entferne die Konstante:

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

= \frac{1}{k ^{2}} \cdot \int \frac{u - 1}{u} d u

Multipliziere aus

\frac{u - 1}{u} : 1 - \frac{1}{u}

= \frac{1}{k ^{2}} \cdot \int 1 - \frac{1}{u} d u

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{k ^{2}} (\int 1 d u - \int \frac{1}{u} d u)

\int 1 d u = u

\int \frac{1}{u} d u = ln ∣ u ∣

= \frac{1}{k ^{2}} (u - ln ∣ u ∣)

Setze in

u = 1 - k v

ein

= \frac{1}{k ^{2}} (1 - k v - ln ∣ 1 - k v ∣)

Füge eine Konstante zur Lösung hinzu

= \frac{1}{k ^{2}} (1 - k v - ln ∣ 1 - k v ∣) + C

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