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Beliebt Rechnen >

integral of 1/(x^8+x^6)

Lösung

\int \frac{1}{x ^{8} + x ^{6}} d x

Lösung

- \frac{1}{x} + \frac{1}{3 x ^{3}} - \frac{1}{5 x ^{5}} - arctan (x) + C

Schritte zur Lösung

\int \frac{1}{x ^{8} + x ^{6}} d x

Ermittle den Partialbruch von

\frac{1}{x ^{8} + x ^{6}} : \frac{1}{x ^{2}} - \frac{1}{x ^{4}} + \frac{1}{x ^{6}} - \frac{1}{x ^{2} + 1}

= \int \frac{1}{x ^{2}} - \frac{1}{x ^{4}} + \frac{1}{x ^{6}} - \frac{1}{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 \frac{1}{x ^{2}} d x - \int \frac{1}{x ^{4}} d x + \int \frac{1}{x ^{6}} d x - \int \frac{1}{x ^{2} + 1} d x

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

\int \frac{1}{x ^{4}} d x = - \frac{1}{3 x ^{3}}

\int \frac{1}{x ^{6}} d x = - \frac{1}{5 x ^{5}}

\int \frac{1}{x ^{2} + 1} d x = arctan (x)

= - \frac{1}{x} - (- \frac{1}{3 x ^{3}}) - \frac{1}{5 x ^{5}} - arctan (x)

Vereinfache

= - \frac{1}{x} + \frac{1}{3 x ^{3}} - \frac{1}{5 x ^{5}} - arctan (x)

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{x} + \frac{1}{3 x ^{3}} - \frac{1}{5 x ^{5}} - arctan (x) + C

Graph

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