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Beliebt Rechnen >

integral of (x^6-x^3+49)/(x^4+7x^2)

Lösung

\int \frac{x ^{6} - x ^{3} + 49}{x ^{4} + 7 x ^{2}} d x

Lösung

\frac{x ^{3}}{3} - 7 x - \frac{7}{x} - \frac{1}{2} ln \frac{x ^{2}}{7} + 1 + 67 arctan (\frac{x}{7}) + C

Schritte zur Lösung

\int \frac{x ^{6} - x ^{3} + 49}{x ^{4} + 7 x ^{2}} d x

Ermittle den Partialbruch von

\frac{x ^{6} - x ^{3} + 49}{x ^{4} + 7 x ^{2}} : x^{2} - 7 + \frac{7}{x ^{2}} + \frac{- x + 42}{x ^{2} + 7}

= \int x^{2} - 7 + \frac{7}{x ^{2}} + \frac{- x + 42}{x ^{2} + 7} 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 7 d x + \int \frac{7}{x ^{2}} d x + \int \frac{- x + 42}{x ^{2} + 7} d x

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

\int 7 d x = 7 x

\int \frac{7}{x ^{2}} d x = - \frac{7}{x}

\int \frac{- x + 42}{x ^{2} + 7} d x = - \frac{1}{2} ln \frac{x ^{2}}{7} + 1 + 67 arctan (\frac{x}{7})

= \frac{x ^{3}}{3} - 7 x - \frac{7}{x} - \frac{1}{2} ln \frac{x ^{2}}{7} + 1 + 67 arctan (\frac{x}{7})

Füge eine Konstante zur Lösung hinzu

= \frac{x ^{3}}{3} - 7 x - \frac{7}{x} - \frac{1}{2} ln \frac{x ^{2}}{7} + 1 + 67 arctan (\frac{x}{7}) + C

Graph

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