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Beliebt Rechnen >

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

Lösung

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

Lösung

\frac{x ^{2}}{2} - x + 6 ln ∣ x + 1 ∣ - 10 ln ∣ x + 2 ∣ - 2 ln ∣ 2 x + 4 ∣ + 2 ln ∣ 2 x + 2 ∣ + C

Schritte zur Lösung

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

Multipliziere aus

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

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

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

\int \frac{5 x}{x ^{2} + 3 x + 2} d x = 5 (- ln ∣ x + 1 ∣ + 2 ln ∣ x + 2 ∣)

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

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

Vereinfache

\frac{x ^{2}}{2} - 3 x - ln ∣ x + 1 ∣ + 8 ln ∣ x + 2 ∣ + 2 (ln ∣ x + 1 ∣ - 4 ln ∣ x + 2 ∣ + x) - 5 (- ln ∣ x + 1 ∣ + 2 ln ∣ x + 2 ∣) - 4 (\frac{1}{2} ln ∣ 2 x + 4 ∣ - \frac{1}{2} ln ∣ 2 x + 2 ∣) : \frac{x ^{2}}{2} - x + 6 ln ∣ x + 1 ∣ - 10 ln ∣ x + 2 ∣ - 2 ln ∣ 2 x + 4 ∣ + 2 ln ∣ 2 x + 2 ∣

= \frac{x ^{2}}{2} - x + 6 ln ∣ x + 1 ∣ - 10 ln ∣ x + 2 ∣ - 2 ln ∣ 2 x + 4 ∣ + 2 ln ∣ 2 x + 2 ∣

Füge eine Konstante zur Lösung hinzu

= \frac{x ^{2}}{2} - x + 6 ln ∣ x + 1 ∣ - 10 ln ∣ x + 2 ∣ - 2 ln ∣ 2 x + 4 ∣ + 2 ln ∣ 2 x + 2 ∣ + C

Graph

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