What is the integral of (x-1)^{-1} ?

The integral of (x-1)^{-1} is ln|x-1|+C

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integral of (x-1)^{-1}

\int (x - 1)^{- 1} d x

ln ∣ x - 1 ∣ + C

Solution steps

\int (x - 1)^{- 1} d x

Apply exponent rule:

a^{- 1} = \frac{1}{a}

= \int \frac{1}{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