What is the integral of (x-2)/((x-1)(x+2)) ?

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

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integral of (x-2)/((x-1)(x+2))

\int \frac{x - 2}{( x - 1 ) ( x + 2 )} d x

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

Solution steps

\int \frac{x - 2}{( x - 1 ) ( x + 2 )} d x

Take the partial fraction of

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

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

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

\int \frac{4}{3 ( x + 2 )} d x = \frac{4}{3} ln ∣ x + 2 ∣

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

Add a constant to the solution

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

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