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

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

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

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

- ln ∣ 1 - x ∣ + C

Solution steps

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

Apply exponent rule:

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

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

Apply u-substitution

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

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= - ln ∣ u ∣

Substitute back

u = 1 - x

= - ln ∣ 1 - x ∣

Add a constant to the solution

= - ln ∣ 1 - x ∣ + C

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