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

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

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

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

ln ∣ x ∣ - ln ∣ x - 1 ∣ - \frac{2}{x} + C

Solution steps

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

Expand

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

\int \frac{x}{x ^{3} - x ^{2}} d x = - ln ∣ x ∣ + ln ∣ x - 1 ∣

\int \frac{2}{x ^{3} - x ^{2}} d x = 2 (- ln ∣ x ∣ + \frac{1}{x} + ln ∣ x - 1 ∣)

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

Simplify

- ln ∣ x ∣ + ln ∣ x - 1 ∣ - 2 (- ln ∣ x ∣ + \frac{1}{x} + ln ∣ x - 1 ∣) : ln ∣ x ∣ - ln ∣ x - 1 ∣ - \frac{2}{x}

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

Add a constant to the solution

= ln ∣ x ∣ - ln ∣ x - 1 ∣ - \frac{2}{x} + C

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