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

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

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integral of x/(x^2-x)

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

ln ∣ x - 1 ∣ + C

Solution steps

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

Factor

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

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

Apply u-substitution

= \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= ln ∣ u ∣

Substitute back

u = x - 1

= ln ∣ x - 1 ∣

Add a constant to the solution

= ln ∣ x - 1 ∣ + C