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Beliebt Rechnen >

integral of (x^3+2x^2+5x+12)/(x+2)

Lösung

\int \frac{x ^{3} + 2 x ^{2} + 5 x + 12}{x + 2} d x

Lösung

- 2 x^{2} + x + \frac{1}{3} (x + 2)^{3} + 2 ln ∣ x + 2 ∣ + 10 + C

Schritte zur Lösung

\int \frac{x ^{3} + 2 x ^{2} + 5 x + 12}{x + 2} d x

Multipliziere aus

\frac{x ^{3} + 2 x ^{2} + 5 x + 12}{x + 2} : \frac{x ^{3}}{x + 2} + \frac{2 x ^{2}}{x + 2} + \frac{5 x}{x + 2} + \frac{12}{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 ^{3}}{x + 2} d x + \int \frac{2 x ^{2}}{x + 2} d x + \int \frac{5 x}{x + 2} d x + \int \frac{12}{x + 2} d x

\int \frac{x ^{3}}{x + 2} d x = - 3 x^{2} + \frac{1}{3} (x + 2)^{3} - 8 ln ∣ x + 2 ∣ + 12

\int \frac{2 x ^{2}}{x + 2} d x = x^{2} - 4 x + 8 ln ∣ x + 2 ∣ - 12

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

\int \frac{12}{x + 2} d x = 12 ln ∣ x + 2 ∣

= - 3 x^{2} + \frac{1}{3} (x + 2)^{3} - 8 ln ∣ x + 2 ∣ + 12 + x^{2} - 4 x + 8 ln ∣ x + 2 ∣ - 12 + 5 (x + 2 - 2 ln ∣ x + 2 ∣) + 12 ln ∣ x + 2 ∣

Vereinfache

- 3 x^{2} + \frac{1}{3} (x + 2)^{3} - 8 ln ∣ x + 2 ∣ + 12 + x^{2} - 4 x + 8 ln ∣ x + 2 ∣ - 12 + 5 (x + 2 - 2 ln ∣ x + 2 ∣) + 12 ln ∣ x + 2 ∣ : - 2 x^{2} + x + \frac{1}{3} (x + 2)^{3} + 2 ln ∣ x + 2 ∣ + 10

= - 2 x^{2} + x + \frac{1}{3} (x + 2)^{3} + 2 ln ∣ x + 2 ∣ + 10

Füge eine Konstante zur Lösung hinzu

= - 2 x^{2} + x + \frac{1}{3} (x + 2)^{3} + 2 ln ∣ x + 2 ∣ + 10 + C

Graph

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