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

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

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

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

- ln ∣ x - 1 ∣ + 3 ln ∣ x ∣ - \frac{1}{x} + C

Solution steps

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

Expand

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

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

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

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

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

Simplify

2 ln ∣ x - 1 ∣ - 2 (- ln ∣ x ∣ + ln ∣ x - 1 ∣) - (- ln ∣ x ∣ + \frac{1}{x} + ln ∣ x - 1 ∣) : - ln ∣ x - 1 ∣ + 3 ln ∣ x ∣ - \frac{1}{x}

= - ln ∣ x - 1 ∣ + 3 ln ∣ x ∣ - \frac{1}{x}

Add a constant to the solution

= - ln ∣ x - 1 ∣ + 3 ln ∣ x ∣ - \frac{1}{x} + C

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