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Beliebt Rechnen >

integral of (sqrt(x))/(1-x)

Lösung

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

Lösung

ln x + 1 - ln x - 1 - 2 x + C

Schritte zur Lösung

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

Wende U-Substitution an

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

Entferne die Konstante:

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

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

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

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

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

\int \frac{1}{1 - u ^{2}} d u = \frac{ln ∣ u + 1 ∣}{2} - \frac{ln ∣ u - 1 ∣}{2}

\int 1 d u = u

= 2 (\frac{ln ∣ u + 1 ∣}{2} - \frac{ln ∣ u - 1 ∣}{2} - u)

Setze in

u = x

ein

= 2 (\frac{ln ∣ x + 1 ∣}{2} - \frac{ln ∣ x - 1 ∣}{2} - x)

Vereinfache

2 (\frac{ln ∣ x + 1 ∣}{2} - \frac{ln ∣ x - 1 ∣}{2} - x) : ln x + 1 - ln x - 1 - 2 x

= ln x + 1 - ln x - 1 - 2 x

Füge eine Konstante zur Lösung hinzu

= ln x + 1 - ln x - 1 - 2 x + C

Graph

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