What is the integral of 1/(5-x) ?

The integral of 1/(5-x) is -ln|5-x|+C

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integral of 1/(5-x)

\int \frac{1}{5 - x} d x

- ln ∣ 5 - x ∣ + C

Solution steps

\int \frac{1}{5 - 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 = 5 - x

= - ln ∣ 5 - x ∣

Add a constant to the solution

= - ln ∣ 5 - x ∣ + C