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

Lösung

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

\frac{x ^{2}}{2} - 2 x + \frac{1}{3} ln ∣ x - 1 ∣ + \frac{14}{3} ln ∣ x + 2 ∣ + C

Schritte zur Lösung

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

Multipliziere aus

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

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

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

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

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

Vereinfache

\frac{x ^{2}}{2} - x + \frac{1}{3} ln ∣ x - 1 ∣ + \frac{8}{3} ln ∣ x + 2 ∣ - (x + \frac{1}{3} ln ∣ x - 1 ∣ - \frac{4}{3} ln ∣ x + 2 ∣) + \frac{1}{3} ln ∣ x - 1 ∣ + \frac{2}{3} ln ∣ x + 2 ∣ : \frac{x ^{2}}{2} - 2 x + \frac{1}{3} ln ∣ x - 1 ∣ + \frac{14}{3} ln ∣ x + 2 ∣

= \frac{x ^{2}}{2} - 2 x + \frac{1}{3} ln ∣ x - 1 ∣ + \frac{14}{3} ln ∣ x + 2 ∣

Füge eine Konstante zur Lösung hinzu

= \frac{x ^{2}}{2} - 2 x + \frac{1}{3} ln ∣ x - 1 ∣ + \frac{14}{3} ln ∣ x + 2 ∣ + C

\int \frac{x ^{2} + x + 1}{2 x ^{3} + x ^{2} + 2 x + 1} d x

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

\int (x^{2} + 4) cos (5 x) d x

\int \frac{sin ( x )}{3 cos ( x )} d x

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