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

Lösung

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

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

Schritte zur Lösung

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

\frac{x ^{2} + 2}{x ^{2} - 1} = \frac{3}{x ^{2} - 1} + 1

= \int \frac{3}{x ^{2} - 1} + 1 d x

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{3}{x ^{2} - 1} d x + \int 1 d x

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

\int 1 d x = x

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

Füge eine Konstante zur Lösung hinzu

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

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