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

x \to - 2 lim (\frac{3 x ^{2} + a x + a + 9}{x ^{2} + x - 2})

\frac{5 s e ^{- 2 s}}{2 s ^{2} + 50}

\int \frac{4 x ^{2} - 12 x - 19}{( x + 1 ) ^{2} ( x - 2 )} d x

\frac{d y}{d x} (x + 3) = y

\int e^{2 y} d y

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