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Beliebt Rechnen >

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

Lösung

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

Lösung

6 (\frac{1}{4} ln 4 x^{2} + 12 x - 40 + \frac{3}{28} (ln \frac{10}{7} + \frac{2}{7} x - ln - \frac{4}{7} + \frac{2}{7} x)) + 2 (- \frac{1}{7} ln ∣ x + 5 ∣ + \frac{1}{7} ln ∣ x - 2 ∣) + C

Schritte zur Lösung

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

Multipliziere aus

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

\int \frac{3 x}{( x + 5 ) ( x - 2 )} d x = 6 (\frac{1}{4} ln 4 x^{2} + 12 x - 40 + \frac{3}{28} (ln \frac{10}{7} + \frac{2}{7} x - ln - \frac{4}{7} + \frac{2}{7} x))

\int \frac{2}{( x + 5 ) ( x - 2 )} d x = 2 (- \frac{1}{7} ln ∣ x + 5 ∣ + \frac{1}{7} ln ∣ x - 2 ∣)

= 6 (\frac{1}{4} ln 4 x^{2} + 12 x - 40 + \frac{3}{28} (ln \frac{10}{7} + \frac{2}{7} x - ln - \frac{4}{7} + \frac{2}{7} x)) + 2 (- \frac{1}{7} ln ∣ x + 5 ∣ + \frac{1}{7} ln ∣ x - 2 ∣)

Füge eine Konstante zur Lösung hinzu

= 6 (\frac{1}{4} ln 4 x^{2} + 12 x - 40 + \frac{3}{28} (ln \frac{10}{7} + \frac{2}{7} x - ln - \frac{4}{7} + \frac{2}{7} x)) + 2 (- \frac{1}{7} ln ∣ x + 5 ∣ + \frac{1}{7} ln ∣ x - 2 ∣) + C

Graph

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