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

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

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

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

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

Solution steps

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

Expand

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

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

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

Simplify

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

Add a constant to the solution

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

\int 0.5 e^{- 0.5 (x - 1)} d x

\frac{d}{d x} (\frac{1}{x ^{4} + x ^{2} + 1})

x \to - 2 lim (x + 2)

n = 0 \sum \infty \frac{3}{3 ^{n}}

\frac{x ^{2}}{x ^{2} - 9}

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