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

The integral of (x-1)/(x-5) is x-5+4ln|x-5|+C

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integral of (x-1)/(x-5)

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

x - 5 + 4 ln ∣ x - 5 ∣ + C

Solution steps

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

Expand

\frac{x - 1}{x - 5} : \frac{x}{x - 5} - \frac{1}{x - 5}

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

\int \frac{x}{x - 5} d x = x - 5 + 5 ln ∣ x - 5 ∣

\int \frac{1}{x - 5} d x = ln ∣ x - 5 ∣

= x - 5 + 5 ln ∣ x - 5 ∣ - ln ∣ x - 5 ∣

Simplify

= x - 5 + 4 ln ∣ x - 5 ∣

Add a constant to the solution

= x - 5 + 4 ln ∣ x - 5 ∣ + C

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What is the integral of (x-1)/(x-5) ?
The integral of (x-1)/(x-5) is x-5+4ln|x-5|+C