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

Lösung

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

\frac{1}{2} (x - 2)^{2} + 4 x - 8 + 3 ln ∣ x - 2 ∣ + C

Schritte zur Lösung

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

Multipliziere aus

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

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

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

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

Vereinfache

\frac{( x - 2 ) ^{2}}{2} + 4 (x - 2) + 4 ln ∣ x - 2 ∣ - ln ∣ x - 2 ∣ : \frac{1}{2} (x - 2)^{2} + 4 x - 8 + 3 ln ∣ x - 2 ∣

= \frac{1}{2} (x - 2)^{2} + 4 x - 8 + 3 ln ∣ x - 2 ∣

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} (x - 2)^{2} + 4 x - 8 + 3 ln ∣ x - 2 ∣ + C

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