integral of (x^{14})/(x^5-1)

Lösung

\int \frac{x ^{14}}{x ^{5} - 1} d x

\frac{x ^{10}}{10} + \frac{x ^{5}}{5} + \frac{1}{5} ln x^{5} - 1 + C

Schritte zur Lösung

\int \frac{x ^{14}}{x ^{5} - 1} d x

Lange Division

\frac{x ^{14}}{x ^{5} - 1} : x^{9} + x^{4} + \frac{x ^{4}}{x ^{5} - 1}

= \int x^{9} + x^{4} + \frac{x ^{4}}{x ^{5} - 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^{9} d x + \int x^{4} d x + \int \frac{x ^{4}}{x ^{5} - 1} d x

\int x^{9} d x = \frac{x ^{10}}{10}

\int x^{4} d x = \frac{x ^{5}}{5}

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

= \frac{x ^{10}}{10} + \frac{x ^{5}}{5} + \frac{1}{5} ln x^{5} - 1

Füge eine Konstante zur Lösung hinzu

= \frac{x ^{10}}{10} + \frac{x ^{5}}{5} + \frac{1}{5} ln x^{5} - 1 + C

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