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Beliebt Rechnen >

integral of (x^2-4x+1)/(x-1)

Lösung

\int \frac{x ^{2} - 4 x + 1}{x - 1} d x

Lösung

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

Schritte zur Lösung

\int \frac{x ^{2} - 4 x + 1}{x - 1} d x

Multipliziere aus

\frac{x ^{2} - 4 x + 1}{x - 1} : \frac{x ^{2}}{x - 1} - \frac{4 x}{x - 1} + \frac{1}{x - 1}

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 ^{2}}{x - 1} d x - \int \frac{4 x}{x - 1} d x + \int \frac{1}{x - 1} d x

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

\int \frac{4 x}{x - 1} d x = 4 (x - 1 + ln ∣ x - 1 ∣)

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

= \frac{( x - 1 ) ^{2}}{2} + 2 (x - 1) + ln ∣ x - 1 ∣ - 4 (x - 1 + ln ∣ x - 1 ∣) + ln ∣ x - 1 ∣

Vereinfache

\frac{( x - 1 ) ^{2}}{2} + 2 (x - 1) + ln ∣ x - 1 ∣ - 4 (x - 1 + ln ∣ x - 1 ∣) + ln ∣ x - 1 ∣ : - 2 x + \frac{1}{2} (x - 1)^{2} - 2 ln ∣ x - 1 ∣ + 2

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

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\int x cos (\frac{n x}{2}) d x

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