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

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

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

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

ln ∣ x - 1 ∣ - ln ∣ x - 2 ∣ + C

Solution steps

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

Take the partial fraction of

\frac{x}{x ^{3} - 3 x ^{2} + 2 x} : - \frac{1}{x - 1} + \frac{1}{x - 2}

= - \int - \frac{1}{x - 1} + \frac{1}{x - 2} d x

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

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

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

= - (- ln ∣ x - 1 ∣ + ln ∣ x - 2 ∣)

Simplify

= ln ∣ x - 1 ∣ - ln ∣ x - 2 ∣

Add a constant to the solution

= ln ∣ x - 1 ∣ - ln ∣ x - 2 ∣ + C

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