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Beliebt Rechnen >

integral of (x^5)/(x^3+8)

Lösung

\int \frac{x ^{5}}{x ^{3} + 8} d x

Lösung

\frac{x ^{3}}{3} - \frac{8}{3} ln x^{3} + 8 + C

Schritte zur Lösung

\int \frac{x ^{5}}{x ^{3} + 8} d x

Lange Division

\frac{x ^{5}}{x ^{3} + 8} : x^{2} - \frac{8 x ^{2}}{x ^{3} + 8}

= \int x^{2} - \frac{8 x ^{2}}{x ^{3} + 8} 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^{2} d x - \int \frac{8 x ^{2}}{x ^{3} + 8} d x

\int x^{2} d x = \frac{x ^{3}}{3}

\int \frac{8 x ^{2}}{x ^{3} + 8} d x = \frac{8}{3} ln x^{3} + 8

= \frac{x ^{3}}{3} - \frac{8}{3} ln x^{3} + 8

Füge eine Konstante zur Lösung hinzu

= \frac{x ^{3}}{3} - \frac{8}{3} ln x^{3} + 8 + C

Graph

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