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Beliebt Rechnen >

integral of (x^2+8)/(x^2-5x+6)

Lösung

\int \frac{x ^{2} + 8}{x ^{2} - 5 x + 6} d x

Lösung

4 (\frac{5}{8} ln 4 x^{2} - 20 x + 24 - \frac{13}{8} ln ∣ 2 x - 4 ∣ + \frac{13}{8} ln ∣ 2 x - 6 ∣) + x + 8 (- ln ∣ x - 2 ∣ + ln ∣ x - 3 ∣) + C

Schritte zur Lösung

\int \frac{x ^{2} + 8}{x ^{2} - 5 x + 6} d x

Multipliziere aus

\frac{x ^{2} + 8}{x ^{2} - 5 x + 6} : \frac{x ^{2}}{x ^{2} - 5 x + 6} + \frac{8}{x ^{2} - 5 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{x ^{2}}{x ^{2} - 5 x + 6} d x + \int \frac{8}{x ^{2} - 5 x + 6} d x

\int \frac{x ^{2}}{x ^{2} - 5 x + 6} d x = 4 (\frac{5}{8} ln 4 x^{2} - 20 x + 24 - \frac{13}{8} ln ∣ 2 x - 4 ∣ + \frac{13}{8} ln ∣ 2 x - 6 ∣) + x

\int \frac{8}{x ^{2} - 5 x + 6} d x = 8 (- ln ∣ x - 2 ∣ + ln ∣ x - 3 ∣)

= 4 (\frac{5}{8} ln 4 x^{2} - 20 x + 24 - \frac{13}{8} ln ∣ 2 x - 4 ∣ + \frac{13}{8} ln ∣ 2 x - 6 ∣) + x + 8 (- ln ∣ x - 2 ∣ + ln ∣ x - 3 ∣)

Füge eine Konstante zur Lösung hinzu

= 4 (\frac{5}{8} ln 4 x^{2} - 20 x + 24 - \frac{13}{8} ln ∣ 2 x - 4 ∣ + \frac{13}{8} ln ∣ 2 x - 6 ∣) + x + 8 (- ln ∣ x - 2 ∣ + ln ∣ x - 3 ∣) + C

Graph

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