integral of (x^5)/(x^4-1)

Lösung

\int \frac{x ^{5}}{x ^{4} - 1} d x

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

Schritte zur Lösung

\int \frac{x ^{5}}{x ^{4} - 1} d x

Lange Division

\frac{x ^{5}}{x ^{4} - 1} : x + \frac{x}{x ^{4} - 1}

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

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

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

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

Füge eine Konstante zur Lösung hinzu

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

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