What is the integral of (x-1)/(x-2) ?

The integral of (x-1)/(x-2) is x-2+ln|x-2|+C

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integral of (x-1)/(x-2)

\int \frac{x - 1}{x - 2} d x

x - 2 + ln ∣ x - 2 ∣ + C

Solution steps

\int \frac{x - 1}{x - 2} d x

Expand

\frac{x - 1}{x - 2} : \frac{x}{x - 2} - \frac{1}{x - 2}

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 - 2} d x - \int \frac{1}{x - 2} d x

\int \frac{x}{x - 2} d x = x - 2 + 2 ln ∣ x - 2 ∣

\int \frac{1}{x - 2} d x = ln ∣ x - 2 ∣

= x - 2 + 2 ln ∣ x - 2 ∣ - ln ∣ x - 2 ∣

Simplify

= x - 2 + ln ∣ x - 2 ∣

Add a constant to the solution

= x - 2 + ln ∣ x - 2 ∣ + C

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What is the integral of (x-1)/(x-2) ?
The integral of (x-1)/(x-2) is x-2+ln|x-2|+C