What is the integral of (2x^3-12x+10)/(x^2-x-6) ?

The integral of (2x^3-12x+10)/(x^2-x-6) is

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integral of (2x^3-12x+10)/(x^2-x-6)

\int \frac{2 x ^{3} - 12 x + 10}{x ^{2} - x - 6} d x

x^{2} + 2 x - \frac{7}{5} ln \frac{2}{5} x + \frac{4}{5} + \frac{7}{5} ln \frac{2}{5} x - \frac{6}{5} - \frac{22}{5} ln ∣ x + 2 ∣ + \frac{22}{5} ln ∣ x - 3 ∣ + ln 4 x^{2} - 4 x - 24 + \frac{6}{5} (ln \frac{2}{5} x + \frac{4}{5} - ln \frac{2}{5} x - \frac{6}{5}) + C

Solution steps

\int \frac{2 x ^{3} - 12 x + 10}{x ^{2} - x - 6} d x

Expand

\frac{2 x ^{3} - 12 x + 10}{x ^{2} - x - 6} : \frac{2 x ^{3}}{x ^{2} - x - 6} - \frac{12 x}{x ^{2} - x - 6} + \frac{10}{x ^{2} - x - 6}

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{2 x ^{3}}{x ^{2} - x - 6} d x - \int \frac{12 x}{x ^{2} - x - 6} d x + \int \frac{10}{x ^{2} - x - 6} d x

\int \frac{2 x ^{3}}{x ^{2} - x - 6} d x = x^{2} + 2 x + 7 ln 4 x^{2} - 4 x - 24 - \frac{7}{5} (ln \frac{2}{5} x + \frac{4}{5} - ln \frac{2}{5} x - \frac{6}{5}) - \frac{12}{5} ln ∣ x + 2 ∣ + \frac{12}{5} ln ∣ x - 3 ∣

\int \frac{12 x}{x ^{2} - x - 6} d x = 24 (\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))

\int \frac{10}{x ^{2} - x - 6} d x = 10 (- \frac{1}{5} ln ∣ x + 2 ∣ + \frac{1}{5} ln ∣ x - 3 ∣)

= x^{2} + 2 x + 7 ln 4 x^{2} - 4 x - 24 - \frac{7}{5} (ln \frac{2}{5} x + \frac{4}{5} - ln \frac{2}{5} x - \frac{6}{5}) - \frac{12}{5} ln ∣ x + 2 ∣ + \frac{12}{5} ln ∣ x - 3 ∣ - 24 (\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)) + 10 (- \frac{1}{5} ln ∣ x + 2 ∣ + \frac{1}{5} ln ∣ x - 3 ∣)

Simplify

x^{2} + 2 x + 7 ln 4 x^{2} - 4 x - 24 - \frac{7}{5} (ln \frac{2}{5} x + \frac{4}{5} - ln \frac{2}{5} x - \frac{6}{5}) - \frac{12}{5} ln ∣ x + 2 ∣ + \frac{12}{5} ln ∣ x - 3 ∣ - 24 (\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)) + 10 (- \frac{1}{5} ln ∣ x + 2 ∣ + \frac{1}{5} ln ∣ x - 3 ∣) : x^{2} + 2 x - \frac{7}{5} ln \frac{2}{5} x + \frac{4}{5} + \frac{7}{5} ln \frac{2}{5} x - \frac{6}{5} - \frac{22}{5} ln ∣ x + 2 ∣ + \frac{22}{5} ln ∣ x - 3 ∣ + ln 4 x^{2} - 4 x - 24 + \frac{6}{5} (ln \frac{2}{5} x + \frac{4}{5} - ln \frac{2}{5} x - \frac{6}{5})

= x^{2} + 2 x - \frac{7}{5} ln \frac{2}{5} x + \frac{4}{5} + \frac{7}{5} ln \frac{2}{5} x - \frac{6}{5} - \frac{22}{5} ln ∣ x + 2 ∣ + \frac{22}{5} ln ∣ x - 3 ∣ + ln 4 x^{2} - 4 x - 24 + \frac{6}{5} (ln \frac{2}{5} x + \frac{4}{5} - ln \frac{2}{5} x - \frac{6}{5})

Add a constant to the solution

= x^{2} + 2 x - \frac{7}{5} ln \frac{2}{5} x + \frac{4}{5} + \frac{7}{5} ln \frac{2}{5} x - \frac{6}{5} - \frac{22}{5} ln ∣ x + 2 ∣ + \frac{22}{5} ln ∣ x - 3 ∣ + ln 4 x^{2} - 4 x - 24 + \frac{6}{5} (ln \frac{2}{5} x + \frac{4}{5} - ln \frac{2}{5} x - \frac{6}{5}) + C

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What is the integral of (2x^3-12x+10)/(x^2-x-6) ?
The integral of (2x^3-12x+10)/(x^2-x-6) is