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

The integral of (x-2)/(x^2-x) is 2ln|x|-ln|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 ∣ - 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{1}{x - 1}

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

\int \frac{2}{x} d x = 2 ln ∣ x ∣

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

= 2 ln ∣ x ∣ - ln ∣ x - 1 ∣

Add a constant to the solution

= 2 ln ∣ x ∣ - ln ∣ x - 1 ∣ + C

\int_{0}^{2} 2^{x} d x

\int \frac{1}{3 x ^{2} + 1} d x

y d u + (2 y - u) d y = 0

\int \frac{1}{4 x + 2} d x

k = 0 \sum \infty (2 k + 1) x^{2 k}

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