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

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

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

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

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

Solution steps

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

Expand

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

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

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

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

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

Simplify

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

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

Add a constant to the solution

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

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