integral of (6x^2-11x+7)/(3x-1)

Lösung

\int \frac{6 x ^{2} - 11 x + 7}{3 x - 1} d x

x^{2} - 3 x + \frac{4}{3} ln ∣ 3 x - 1 ∣ - \frac{1}{3} + \frac{11}{9} + C

Schritte zur Lösung

\int \frac{6 x ^{2} - 11 x + 7}{3 x - 1} d x

Multipliziere aus

\frac{6 x ^{2} - 11 x + 7}{3 x - 1} : \frac{6 x ^{2}}{3 x - 1} - \frac{11 x}{3 x - 1} + \frac{7}{3 x - 1}

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{6 x ^{2}}{3 x - 1} d x - \int \frac{11 x}{3 x - 1} d x + \int \frac{7}{3 x - 1} d x

\int \frac{6 x ^{2}}{3 x - 1} d x = \frac{1}{9} (9 x^{2} + 6 x + 2 ln ∣ 3 x - 1 ∣ - 3)

\int \frac{11 x}{3 x - 1} d x = \frac{11}{9} (3 x - 1 + ln ∣ 3 x - 1 ∣)

\int \frac{7}{3 x - 1} d x = \frac{7}{3} ln ∣ 3 x - 1 ∣

= \frac{1}{9} (9 x^{2} + 6 x + 2 ln ∣ 3 x - 1 ∣ - 3) - \frac{11}{9} (3 x - 1 + ln ∣ 3 x - 1 ∣) + \frac{7}{3} ln ∣ 3 x - 1 ∣

Vereinfache

\frac{1}{9} (9 x^{2} + 6 x + 2 ln ∣ 3 x - 1 ∣ - 3) - \frac{11}{9} (3 x - 1 + ln ∣ 3 x - 1 ∣) + \frac{7}{3} ln ∣ 3 x - 1 ∣ : x^{2} - 3 x + \frac{4}{3} ln ∣ 3 x - 1 ∣ - \frac{1}{3} + \frac{11}{9}

= x^{2} - 3 x + \frac{4}{3} ln ∣ 3 x - 1 ∣ - \frac{1}{3} + \frac{11}{9}

Füge eine Konstante zur Lösung hinzu

= x^{2} - 3 x + \frac{4}{3} ln ∣ 3 x - 1 ∣ - \frac{1}{3} + \frac{11}{9} + C

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