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

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

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

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

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

Solution steps

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

Expand

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

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

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

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

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

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

Simplify

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

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

Add a constant to the solution

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

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