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Beliebt Rechnen >

integral of (7x-1)/(x^2+x-2)

Lösung

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

Lösung

7 (\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{7 x - 1}{x ^{2} + x - 2} d x

Multipliziere aus

\frac{7 x - 1}{x ^{2} + x - 2} : \frac{7 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{7 x}{x ^{2} + x - 2} d x - \int \frac{1}{x ^{2} + x - 2} d x

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

= 7 (\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))

Vereinfache

7 (\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)) : 7 (\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)

= 7 (\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

= 7 (\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 (-7sqrt(x^2-9))/(x^3)

\int \frac{- 7 x ^{2} - 9}{x ^{3}} d x

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\int cos (t) csc (t) d t

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\int t e^{7 t^{2}} d t