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Beliebt Rechnen >

integral of (x^2+44x-15)/(2x^3+6x)

Lösung

\int \frac{x ^{2} + 44 x - 15}{2 x ^{3} + 6 x} d x

Lösung

- \frac{5}{2} ln ∣ x ∣ + \frac{3}{2} ln x^{2} + 3 + \frac{22}{3} arctan (\frac{x}{3}) + C

Schritte zur Lösung

\int \frac{x ^{2} + 44 x - 15}{2 x ^{3} + 6 x} d x

Ermittle den Partialbruch von

\frac{x ^{2} + 44 x - 15}{2 x ^{3} + 6 x} : - \frac{5}{2 x} + \frac{3 x + 22}{x ^{2} + 3}

= \int - \frac{5}{2 x} + \frac{3 x + 22}{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{5}{2 x} d x + \int \frac{3 x + 22}{x ^{2} + 3} d x

\int \frac{5}{2 x} d x = \frac{5}{2} ln ∣ x ∣

\int \frac{3 x + 22}{x ^{2} + 3} d x = \frac{3}{2} ln x^{2} + 3 + \frac{22}{3} arctan (\frac{x}{3})

= - \frac{5}{2} ln ∣ x ∣ + \frac{3}{2} ln x^{2} + 3 + \frac{22}{3} arctan (\frac{x}{3})

Füge eine Konstante zur Lösung hinzu

= - \frac{5}{2} ln ∣ x ∣ + \frac{3}{2} ln x^{2} + 3 + \frac{22}{3} arctan (\frac{x}{3}) + C

Graph

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