What is the integral of (3x-2)/(x-1) ?

The integral of (3x-2)/(x-1) is 3x-3+ln|x-1|+C

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integral of (3x-2)/(x-1)

\int \frac{3 x - 2}{x - 1} d x

3 x - 3 + ln ∣ x - 1 ∣ + C

Solution steps

\int \frac{3 x - 2}{x - 1} d x

Expand

\frac{3 x - 2}{x - 1} : \frac{3 x}{x - 1} - \frac{2}{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{3 x}{x - 1} d x - \int \frac{2}{x - 1} d x

\int \frac{3 x}{x - 1} d x = 3 (x - 1 + ln ∣ x - 1 ∣)

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

= 3 (x - 1 + ln ∣ x - 1 ∣) - 2 ln ∣ x - 1 ∣

Simplify

3 (x - 1 + ln ∣ x - 1 ∣) - 2 ln ∣ x - 1 ∣ : 3 x - 3 + ln ∣ x - 1 ∣

= 3 x - 3 + ln ∣ x - 1 ∣

Add a constant to the solution

= 3 x - 3 + ln ∣ x - 1 ∣ + C

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What is the integral of (3x-2)/(x-1) ?
The integral of (3x-2)/(x-1) is 3x-3+ln|x-1|+C