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Beliebt Rechnen >

integral of (x-2)/(x^2-4x+4)

Lösung

\int \frac{x - 2}{x ^{2} - 4 x + 4} d x

Lösung

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

Schritte zur Lösung

\int \frac{x - 2}{x ^{2} - 4 x + 4} d x

Multipliziere aus

\frac{x - 2}{x ^{2} - 4 x + 4} : \frac{x}{x ^{2} - 4 x + 4} - \frac{2}{x ^{2} - 4 x + 4}

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{x}{x ^{2} - 4 x + 4} d x - \int \frac{2}{x ^{2} - 4 x + 4} d x

\int \frac{x}{x ^{2} - 4 x + 4} d x = - \frac{x}{x - 2} + ln ∣ x - 2 ∣

\int \frac{2}{x ^{2} - 4 x + 4} d x = - \frac{2}{x - 2}

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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