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Beliebt Rechnen >

integral of (x^6-x^3+64)/(x^4+8x^2)

Lösung

\int \frac{x ^{6} - x ^{3} + 64}{x ^{4} + 8 x ^{2}} d x

Lösung

\frac{x ^{3}}{3} - 8 x - \frac{8}{x} + \frac{1}{2} (- \frac{1}{2} ln \frac{x ^{2}}{8} + 1 + 28 arctan (\frac{x}{2 2})) + C

Schritte zur Lösung

\int \frac{x ^{6} - x ^{3} + 64}{x ^{4} + 8 x ^{2}} d x

Ermittle den Partialbruch von

\frac{x ^{6} - x ^{3} + 64}{x ^{4} + 8 x ^{2}} : x^{2} - 8 + \frac{8}{x ^{2}} + \frac{- x + 56}{x ^{2} + 8}

= \int x^{2} - 8 + \frac{8}{x ^{2}} + \frac{- x + 56}{x ^{2} + 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 8 d x + \int \frac{8}{x ^{2}} d x + \int \frac{- x + 56}{x ^{2} + 8} d x

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

\int 8 d x = 8 x

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

\int \frac{- x + 56}{x ^{2} + 8} d x = \frac{1}{2} (- \frac{1}{2} ln \frac{x ^{2}}{8} + 1 + 28 arctan (\frac{x}{2 2}))

= \frac{x ^{3}}{3} - 8 x - \frac{8}{x} + \frac{1}{2} (- \frac{1}{2} ln \frac{x ^{2}}{8} + 1 + 28 arctan (\frac{x}{2 2}))

Füge eine Konstante zur Lösung hinzu

= \frac{x ^{3}}{3} - 8 x - \frac{8}{x} + \frac{1}{2} (- \frac{1}{2} ln \frac{x ^{2}}{8} + 1 + 28 arctan (\frac{x}{2 2})) + C

Graph

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\frac{d}{d x} (- arctan (x) + 2 arccot (x) - ln (x))

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\int \frac{6}{x ln ( 2 x )} d x