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Beliebt Rechnen >

integral of (x^3+x^2+2x+3)/(x^4+5x^2+6)

Lösung

\int \frac{x ^{3} + x ^{2} + 2 x + 3}{x ^{4} + 5 x ^{2} + 6} d x

Lösung

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

Schritte zur Lösung

\int \frac{x ^{3} + x ^{2} + 2 x + 3}{x ^{4} + 5 x ^{2} + 6} d x

Ermittle den Partialbruch von

\frac{x ^{3} + x ^{2} + 2 x + 3}{x ^{4} + 5 x ^{2} + 6} : \frac{1}{x ^{2} + 2} + \frac{x}{x ^{2} + 3}

= \int \frac{1}{x ^{2} + 2} + \frac{x}{x ^{2} + 3} 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} + 2} d x + \int \frac{x}{x ^{2} + 3} d x

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

\int \frac{x}{x ^{2} + 3} d x = \frac{1}{2} ln x^{2} + 3

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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