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Beliebt Rechnen >

integral of (5x+1)/(x^2+x-2)

Lösung

\int \frac{5 x + 1}{x ^{2} + x - 2} d x

Lösung

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

Schritte zur Lösung

\int \frac{5 x + 1}{x ^{2} + x - 2} d x

Multipliziere aus

\frac{5 x + 1}{x ^{2} + x - 2} : \frac{5 x}{x ^{2} + x - 2} + \frac{1}{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{5 x}{x ^{2} + x - 2} d x + \int \frac{1}{x ^{2} + x - 2} d x

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

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

integral of (x-1)/(x^2+5x+4)

\int \frac{x - 1}{x ^{2} + 5 x + 4} d x

integral of cos(x)cos^2(4x)

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

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\int e^{x} (sin (x) - cos (x)) d x

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\int cos (x) (6 sin^{2} (x) - 1) d x

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

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