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Beliebt Rechnen >

integral of (1-u)/(1+u)

Lösung

\int \frac{1 - u}{1 + u} d u

Lösung

- 1 - u + 2 ln ∣ 1 + u ∣ + C

Schritte zur Lösung

\int \frac{1 - u}{1 + u} d u

Wende U-Substitution an

= \int \frac{- v + 2}{v} d v

Multipliziere aus

\frac{- v + 2}{v} : - 1 + \frac{2}{v}

= \int - 1 + \frac{2}{v} 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 1 d v + \int \frac{2}{v} d v

\int 1 d v = v

\int \frac{2}{v} d v = 2 ln ∣ v ∣

= - v + 2 ln ∣ v ∣

Setze in

v = 1 + u

ein

= - (1 + u) + 2 ln ∣ 1 + u ∣

Vereinfache

= - 1 - u + 2 ln ∣ 1 + u ∣

Füge eine Konstante zur Lösung hinzu

= - 1 - u + 2 ln ∣ 1 + u ∣ + C

Graph

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