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Beliebt Rechnen >

integral of 1/(x^5+16x^3+64x)

Lösung

\int \frac{1}{x ^{5} + 16 x ^{3} + 64 x} d x

Lösung

\frac{1}{64} ln ∣ x ∣ - \frac{1}{128} ln x^{2} + 8 + \frac{1}{16 ( x ^{2} + 8 )} + C

Schritte zur Lösung

\int \frac{1}{x ^{5} + 16 x ^{3} + 64 x} d x

Ermittle den Partialbruch von

\frac{1}{x ^{5} + 16 x ^{3} + 64 x} : \frac{1}{64 x} - \frac{x}{64 ( x ^{2} + 8 )} - \frac{x}{8 ( x ^{2} + 8 ) ^{2}}

= \int \frac{1}{64 x} - \frac{x}{64 ( x ^{2} + 8 )} - \frac{x}{8 ( x ^{2} + 8 ) ^{2}} 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}{64 x} d x - \int \frac{x}{64 ( x ^{2} + 8 )} d x - \int \frac{x}{8 ( x ^{2} + 8 ) ^{2}} d x

\int \frac{1}{64 x} d x = \frac{1}{64} ln ∣ x ∣

\int \frac{x}{64 ( x ^{2} + 8 )} d x = \frac{1}{128} ln x^{2} + 8

\int \frac{x}{8 ( x ^{2} + 8 ) ^{2}} d x = - \frac{1}{16 ( x ^{2} + 8 )}

= \frac{1}{64} ln ∣ x ∣ - \frac{1}{128} ln x^{2} + 8 - (- \frac{1}{16 ( x ^{2} + 8 )})

Vereinfache

= \frac{1}{64} ln ∣ x ∣ - \frac{1}{128} ln x^{2} + 8 + \frac{1}{16 ( x ^{2} + 8 )}

Füge eine Konstante zur Lösung hinzu

= \frac{1}{64} ln ∣ x ∣ - \frac{1}{128} ln x^{2} + 8 + \frac{1}{16 ( x ^{2} + 8 )} + C

Graph

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