integral of 1/(u-u^2)

Lösung

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

ln ∣ u ∣ - ln ∣ u - 1 ∣ + C

Schritte zur Lösung

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

Ermittle den Partialbruch von

\frac{1}{u - u ^{2}} : \frac{1}{u} - \frac{1}{u - 1}

= \int \frac{1}{u} - \frac{1}{u - 1} 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

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

\int \frac{1}{u} d u = ln ∣ u ∣

\int \frac{1}{u - 1} d u = ln ∣ u - 1 ∣

= ln ∣ u ∣ - ln ∣ u - 1 ∣

Füge eine Konstante zur Lösung hinzu

= ln ∣ u ∣ - ln ∣ u - 1 ∣ + C

\int \frac{3 + 7 x}{1 + x ^{2}} d x

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\int x (2 x - 1)^{2} d x

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