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Beliebt Rechnen >

integral of (1-x)/(1-x^2)

Lösung

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

Lösung

\frac{1}{2} ln ∣ x + 1 ∣ - \frac{1}{2} ln ∣ x - 1 ∣ + \frac{1}{2} ln 1 - x^{2} + C

Schritte zur Lösung

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

Multipliziere aus

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

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

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

\int \frac{x}{1 - x ^{2}} d x = - \frac{1}{2} ln 1 - x^{2}

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} ln ∣ x + 1 ∣ - \frac{1}{2} ln ∣ x - 1 ∣ + \frac{1}{2} ln 1 - x^{2} + C

Graph

Beliebte Beispiele

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\int sin^{2} (x + 4) d x

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