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Beliebt Rechnen >

integral of (x^2+3x+5)/(x^3+8)

Lösung

\int \frac{x ^{2} + 3 x + 5}{x ^{3} + 8} d x

Lösung

\frac{1}{4} ln ∣ x + 2 ∣ + \frac{1}{4} (3 (\frac{1}{2} ln x^{2} - 2 x + 4 + \frac{1}{3} arctan (\frac{1}{3} (x - 1))) + \frac{8}{3} arctan (\frac{1}{3} (x - 1))) + C

Schritte zur Lösung

\int \frac{x ^{2} + 3 x + 5}{x ^{3} + 8} d x

Ermittle den Partialbruch von

\frac{x ^{2} + 3 x + 5}{x ^{3} + 8} : \frac{1}{4 ( x + 2 )} + \frac{3 x + 8}{4 ( x ^{2} - 2 x + 4 )}

= \int \frac{1}{4 ( x + 2 )} + \frac{3 x + 8}{4 ( x ^{2} - 2 x + 4 )} 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}{4 ( x + 2 )} d x + \int \frac{3 x + 8}{4 ( x ^{2} - 2 x + 4 )} d x

\int \frac{1}{4 ( x + 2 )} d x = \frac{1}{4} ln ∣ x + 2 ∣

\int \frac{3 x + 8}{4 ( x ^{2} - 2 x + 4 )} d x = \frac{1}{4} (3 (\frac{1}{2} ln x^{2} - 2 x + 4 + \frac{1}{3} arctan (\frac{1}{3} (x - 1))) + \frac{8}{3} arctan (\frac{1}{3} (x - 1)))

= \frac{1}{4} ln ∣ x + 2 ∣ + \frac{1}{4} (3 (\frac{1}{2} ln x^{2} - 2 x + 4 + \frac{1}{3} arctan (\frac{1}{3} (x - 1))) + \frac{8}{3} arctan (\frac{1}{3} (x - 1)))

Füge eine Konstante zur Lösung hinzu

= \frac{1}{4} ln ∣ x + 2 ∣ + \frac{1}{4} (3 (\frac{1}{2} ln x^{2} - 2 x + 4 + \frac{1}{3} arctan (\frac{1}{3} (x - 1))) + \frac{8}{3} arctan (\frac{1}{3} (x - 1))) + C

Graph

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