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

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

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

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

ln (∣ x ∣ ∣ x - 1 ∣) + C

Solution steps

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

Apply u-substitution

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

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= ln ∣ u ∣

Substitute back

u = x (x - 1)

= ln ∣ x (x - 1) ∣

Simplify

ln ∣ x (x - 1) ∣ : ln (∣ x ∣ ∣ x - 1 ∣)

= ln (∣ x ∣ ∣ x - 1 ∣)

Add a constant to the solution

= ln (∣ x ∣ ∣ x - 1 ∣) + C

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