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Beliebt Rechnen >

integral of (x^4+9)/(x^4+9x^2)

Lösung

\int \frac{x ^{4} + 9}{x ^{4} + 9 x ^{2}} d x

Lösung

x - \frac{1}{x} - \frac{10}{3} arctan (\frac{x}{3}) + C

Schritte zur Lösung

\int \frac{x ^{4} + 9}{x ^{4} + 9 x ^{2}} d x

Ermittle den Partialbruch von

\frac{x ^{4} + 9}{x ^{4} + 9 x ^{2}} : 1 + \frac{1}{x ^{2}} - \frac{10}{x ^{2} + 9}

= \int 1 + \frac{1}{x ^{2}} - \frac{10}{x ^{2} + 9} 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 1 d x + \int \frac{1}{x ^{2}} d x - \int \frac{10}{x ^{2} + 9} d x

\int 1 d x = x

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

\int \frac{10}{x ^{2} + 9} d x = \frac{10}{3} arctan (\frac{x}{3})

= x - \frac{1}{x} - \frac{10}{3} arctan (\frac{x}{3})

Füge eine Konstante zur Lösung hinzu

= x - \frac{1}{x} - \frac{10}{3} arctan (\frac{x}{3}) + C

Graph

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\int 6 tan^{5} (x) sec^{2} (x) d x