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Beliebt Rechnen >

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

Lösung

\int (\frac{x ^{2} - 3 x - 5}{x + 2}) d x

Lösung

- 7 x + 5 ln ∣ x + 2 ∣ + \frac{1}{2} (x + 2)^{2} - 14 + C

Schritte zur Lösung

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

Multipliziere aus

\frac{x ^{2} - 3 x - 5}{x + 2} : \frac{x ^{2}}{x + 2} - \frac{3 x}{x + 2} - \frac{5}{x + 2}

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

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

\int \frac{3 x}{x + 2} d x = 3 (x + 2 - 2 ln ∣ x + 2 ∣)

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

= \frac{( x + 2 ) ^{2}}{2} - 4 (x + 2) + 4 ln ∣ x + 2 ∣ - 3 (x + 2 - 2 ln ∣ x + 2 ∣) - 5 ln ∣ x + 2 ∣

Vereinfache

\frac{( x + 2 ) ^{2}}{2} - 4 (x + 2) + 4 ln ∣ x + 2 ∣ - 3 (x + 2 - 2 ln ∣ x + 2 ∣) - 5 ln ∣ x + 2 ∣ : - 7 x + 5 ln ∣ x + 2 ∣ + \frac{1}{2} (x + 2)^{2} - 14

= - 7 x + 5 ln ∣ x + 2 ∣ + \frac{1}{2} (x + 2)^{2} - 14

Füge eine Konstante zur Lösung hinzu

= - 7 x + 5 ln ∣ x + 2 ∣ + \frac{1}{2} (x + 2)^{2} - 14 + C

Graph

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