integral of ((x^5+2))/((x^2-1))

Lösung

\int \frac{( x ^{5} + 2 )}{( x ^{2} - 1 )} d x

\frac{x ^{4}}{4} + \frac{x ^{2}}{2} + \frac{1}{2} ln x^{2} - 1 - ln ∣ x + 1 ∣ + ln ∣ x - 1 ∣ + C

Schritte zur Lösung

\int \frac{x ^{5} + 2}{x ^{2} - 1} d x

Lange Division

\frac{x ^{5} + 2}{x ^{2} - 1} : x^{3} + x + \frac{x + 2}{x ^{2} - 1}

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

\int x^{3} d x = \frac{x ^{4}}{4}

\int x d x = \frac{x ^{2}}{2}

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

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

Füge eine Konstante zur Lösung hinzu

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

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