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

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

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integral of 1/(a-x)

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

- ln ∣ a - x ∣ + C

Solution steps

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

Apply u-substitution

= \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 = a - x

= - ln ∣ a - x ∣

Add a constant to the solution

= - ln ∣ a - x ∣ + C