integral of (4x^3-2x^2+5)/(x^2-x-6)

Lösung

\int \frac{4 x ^{3} - 2 x ^{2} + 5}{x ^{2} - x - 6} d x

2 x^{2} + 2 x + \frac{8}{5} ln ∣ 2 x + 4 ∣ - \frac{29}{5} ln ∣ x + 2 ∣ + \frac{29}{5} ln ∣ x - 3 ∣ - \frac{18}{5} ln ∣ 2 x - 6 ∣ + 14 ln 4 x^{2} - 4 x - 24 - \frac{14}{5} (ln \frac{2}{5} x + \frac{4}{5} - ln \frac{2}{5} x - \frac{6}{5}) + 1 + C

Schritte zur Lösung

\int \frac{4 x ^{3} - 2 x ^{2} + 5}{x ^{2} - x - 6} d x

Multipliziere aus

\frac{4 x ^{3} - 2 x ^{2} + 5}{x ^{2} - x - 6} : \frac{4 x ^{3}}{x ^{2} - x - 6} - \frac{2 x ^{2}}{x ^{2} - x - 6} + \frac{5}{x ^{2} - x - 6}

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{4 x ^{3}}{x ^{2} - x - 6} d x - \int \frac{2 x ^{2}}{x ^{2} - x - 6} d x + \int \frac{5}{x ^{2} - x - 6} d x

\int \frac{4 x ^{3}}{x ^{2} - x - 6} d x = 4 (\frac{x ^{2}}{2} + x + 14 (\frac{1}{4} ln 4 x^{2} - 4 x - 24 - \frac{1}{20} (ln \frac{4}{5} + \frac{2}{5} x - ln - \frac{6}{5} + \frac{2}{5} x)) + 6 (- \frac{1}{5} ln ∣ x + 2 ∣ + \frac{1}{5} ln ∣ x - 3 ∣))

\int \frac{2 x ^{2}}{x ^{2} - x - 6} d x = 2 x - 1 - \frac{8}{5} ln ∣ 2 x + 4 ∣ + \frac{18}{5} ln ∣ 2 x - 6 ∣

\int \frac{5}{x ^{2} - x - 6} d x = - ln ∣ x + 2 ∣ + ln ∣ x - 3 ∣

= 4 (\frac{x ^{2}}{2} + x + 14 (\frac{1}{4} ln 4 x^{2} - 4 x - 24 - \frac{1}{20} (ln \frac{4}{5} + \frac{2}{5} x - ln - \frac{6}{5} + \frac{2}{5} x)) + 6 (- \frac{1}{5} ln ∣ x + 2 ∣ + \frac{1}{5} ln ∣ x - 3 ∣)) - (2 x - 1 - \frac{8}{5} ln ∣ 2 x + 4 ∣ + \frac{18}{5} ln ∣ 2 x - 6 ∣) - ln ∣ x + 2 ∣ + ln ∣ x - 3 ∣

Vereinfache

4 (\frac{x ^{2}}{2} + x + 14 (\frac{1}{4} ln 4 x^{2} - 4 x - 24 - \frac{1}{20} (ln \frac{4}{5} + \frac{2}{5} x - ln - \frac{6}{5} + \frac{2}{5} x)) + 6 (- \frac{1}{5} ln ∣ x + 2 ∣ + \frac{1}{5} ln ∣ x - 3 ∣)) - (2 x - 1 - \frac{8}{5} ln ∣ 2 x + 4 ∣ + \frac{18}{5} ln ∣ 2 x - 6 ∣) - ln ∣ x + 2 ∣ + ln ∣ x - 3 ∣ : 2 x^{2} + 2 x + \frac{8}{5} ln ∣ 2 x + 4 ∣ - \frac{29}{5} ln ∣ x + 2 ∣ + \frac{29}{5} ln ∣ x - 3 ∣ - \frac{18}{5} ln ∣ 2 x - 6 ∣ + 14 ln 4 x^{2} - 4 x - 24 - \frac{14}{5} (ln \frac{2}{5} x + \frac{4}{5} - ln \frac{2}{5} x - \frac{6}{5}) + 1

= 2 x^{2} + 2 x + \frac{8}{5} ln ∣ 2 x + 4 ∣ - \frac{29}{5} ln ∣ x + 2 ∣ + \frac{29}{5} ln ∣ x - 3 ∣ - \frac{18}{5} ln ∣ 2 x - 6 ∣ + 14 ln 4 x^{2} - 4 x - 24 - \frac{14}{5} (ln \frac{2}{5} x + \frac{4}{5} - ln \frac{2}{5} x - \frac{6}{5}) + 1

Füge eine Konstante zur Lösung hinzu

= 2 x^{2} + 2 x + \frac{8}{5} ln ∣ 2 x + 4 ∣ - \frac{29}{5} ln ∣ x + 2 ∣ + \frac{29}{5} ln ∣ x - 3 ∣ - \frac{18}{5} ln ∣ 2 x - 6 ∣ + 14 ln 4 x^{2} - 4 x - 24 - \frac{14}{5} (ln \frac{2}{5} x + \frac{4}{5} - ln \frac{2}{5} x - \frac{6}{5}) + 1 + C

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