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

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

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

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

- ln ∣ x - 1 ∣ + 2 ln ∣ x ∣ + C

Solution steps

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

Expand

\frac{x - 2}{x ( x - 1 )} : \frac{x}{x ( x - 1 )} - \frac{2}{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{x}{x ( x - 1 )} d x - \int \frac{2}{x ( x - 1 )} d x

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

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

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

Simplify

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

= - ln ∣ x - 1 ∣ + 2 ln ∣ x ∣

Add a constant to the solution

= - ln ∣ x - 1 ∣ + 2 ln ∣ x ∣ + C

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