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

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

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

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

- 2 ln ∣ x ∣ + 3 ln ∣ x + 1 ∣ + C

Solution steps

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

Take the partial fraction of

\frac{x - 2}{x ^{2} + x} : - \frac{2}{x} + \frac{3}{x + 1}

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

\int \frac{2}{x} d x = 2 ln ∣ x ∣

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

= - 2 ln ∣ x ∣ + 3 ln ∣ x + 1 ∣

Add a constant to the solution

= - 2 ln ∣ x ∣ + 3 ln ∣ x + 1 ∣ + C

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