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

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

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

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

- x + ln ∣ x - 1 ∣ + 1 + C

Solution steps

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

Expand

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

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

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

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

= 2 ln ∣ x - 1 ∣ - (x - 1 + ln ∣ x - 1 ∣)

Simplify

= - x + ln ∣ x - 1 ∣ + 1

Add a constant to the solution

= - x + ln ∣ x - 1 ∣ + 1 + C

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