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

The integral of (2x^2+x-2)/(x^3-x^2-2x) is ln|x-2|-1/3 (ln| 2/3 x+2/3 |-ln| 2/3 x-4/3 |)+ln|x|+C

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

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

ln ∣ x - 2 ∣ - \frac{1}{3} (ln \frac{2}{3} x + \frac{2}{3} - ln \frac{2}{3} x - \frac{4}{3}) + ln ∣ x ∣ + C

Solution steps

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

Expand

\frac{2 x ^{2} + x - 2}{x ^{3} - x ^{2} - 2 x} : \frac{2 x ^{2}}{x ^{3} - x ^{2} - 2 x} + \frac{x}{x ^{3} - x ^{2} - 2 x} - \frac{2}{x ^{3} - x ^{2} - 2 x}

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

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

\int \frac{x}{x ^{3} - x ^{2} - 2 x} d x = - \frac{1}{3} (ln \frac{2}{3} + \frac{2}{3} x - ln - \frac{4}{3} + \frac{2}{3} x)

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

= 2 (\frac{1}{3} ln ∣ x + 1 ∣ + \frac{2}{3} ln ∣ x - 2 ∣) - \frac{1}{3} (ln \frac{2}{3} + \frac{2}{3} x - ln - \frac{4}{3} + \frac{2}{3} x) - (- ln ∣ x ∣ + \frac{2}{3} ln ∣ x + 1 ∣ + \frac{1}{3} ln ∣ x - 2 ∣)

Simplify

2 (\frac{1}{3} ln ∣ x + 1 ∣ + \frac{2}{3} ln ∣ x - 2 ∣) - \frac{1}{3} (ln \frac{2}{3} + \frac{2}{3} x - ln - \frac{4}{3} + \frac{2}{3} x) - (- ln ∣ x ∣ + \frac{2}{3} ln ∣ x + 1 ∣ + \frac{1}{3} ln ∣ x - 2 ∣) : ln ∣ x - 2 ∣ - \frac{1}{3} (ln \frac{2}{3} x + \frac{2}{3} - ln \frac{2}{3} x - \frac{4}{3}) + ln ∣ x ∣

= ln ∣ x - 2 ∣ - \frac{1}{3} (ln \frac{2}{3} x + \frac{2}{3} - ln \frac{2}{3} x - \frac{4}{3}) + ln ∣ x ∣

Add a constant to the solution

= ln ∣ x - 2 ∣ - \frac{1}{3} (ln \frac{2}{3} x + \frac{2}{3} - ln \frac{2}{3} x - \frac{4}{3}) + ln ∣ x ∣ + C

What is the integral of (2x^2+x-2)/(x^3-x^2-2x) ?
The integral of (2x^2+x-2)/(x^3-x^2-2x) is ln|x-2|-1/3 (ln| 2/3 x+2/3 |-ln| 2/3 x-4/3 |)+ln|x|+C