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Beliebt Rechnen >

integral of (x^3+6x^2+7x+2)/(x^2-x-2)

Lösung

\int \frac{x ^{3} + 6 x ^{2} + 7 x + 2}{x ^{2} - x - 2} d x

Lösung

9 x + 16 ln ∣ x - 2 ∣ + \frac{1}{2} (x - 2)^{2} - 18 + C

Schritte zur Lösung

\int \frac{x ^{3} + 6 x ^{2} + 7 x + 2}{x ^{2} - x - 2} d x

Vereinfache

\frac{x ^{3} + 6 x ^{2} + 7 x + 2}{x ^{2} - x - 2} : \frac{x ^{2} + 5 x + 2}{x - 2}

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

Multipliziere aus

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

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

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

Vereinfache

\frac{( x - 2 ) ^{2}}{2} + 4 (x - 2) + 4 ln ∣ x - 2 ∣ + 5 (x - 2 + 2 ln ∣ x - 2 ∣) + 2 ln ∣ x - 2 ∣ : 9 x + 16 ln ∣ x - 2 ∣ + \frac{1}{2} (x - 2)^{2} - 18

= 9 x + 16 ln ∣ x - 2 ∣ + \frac{1}{2} (x - 2)^{2} - 18

Füge eine Konstante zur Lösung hinzu

= 9 x + 16 ln ∣ x - 2 ∣ + \frac{1}{2} (x - 2)^{2} - 18 + C

Graph

Beliebte Beispiele

integral of x^2ln(2x+5)

\int x^{2} ln (2 x + 5) d x

integral of (7x+2)^{1/2}

\int (7 x + 2)^{\frac{1}{2}} d x

integral of (y^3-y^2+1)

\int (y^{3} - y^{2} + 1) d y

integral of 1/(θ^2)

\int \frac{1}{θ ^{2}} d θ

integral of 1/(a^2x^2-b^2)

\int \frac{1}{a ^{2} x ^{2} - b ^{2}} d x