What is the integral of k/(k-x) ?

The integral of k/(k-x) is -kln|k-x|+C

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integral of k/(k-x)

\int \frac{k}{k - x} d x

- k ln ∣ k - x ∣ + C

Solution steps

\int \frac{k}{k - x} d x

Take the constant out:

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

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

Apply u-substitution

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

Take the constant out:

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

= k (- \int \frac{1}{u} d u)

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= k (- ln ∣ u ∣)

Substitute back

u = k - x

= k (- ln ∣ k - x ∣)

Simplify

= - k ln ∣ k - x ∣

Add a constant to the solution

= - k ln ∣ k - x ∣ + C