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

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

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

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

- ln ∣ x - 1 ∣ + 2 ln ∣ x - 2 ∣ + ln ∣ x ∣ + C

Solution steps

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

Expand

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

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

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

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

= 2 (- ln ∣ x - 1 ∣ + 2 ln ∣ x - 2 ∣) - 3 (- ln ∣ x - 1 ∣ + ln ∣ x - 2 ∣) + ln ∣ x ∣ - 2 ln ∣ x - 1 ∣ + ln ∣ x - 2 ∣

Simplify

2 (- ln ∣ x - 1 ∣ + 2 ln ∣ x - 2 ∣) - 3 (- ln ∣ x - 1 ∣ + ln ∣ x - 2 ∣) + ln ∣ x ∣ - 2 ln ∣ x - 1 ∣ + ln ∣ x - 2 ∣ : - ln ∣ x - 1 ∣ + 2 ln ∣ x - 2 ∣ + ln ∣ x ∣

= - ln ∣ x - 1 ∣ + 2 ln ∣ x - 2 ∣ + ln ∣ x ∣

Add a constant to the solution

= - ln ∣ x - 1 ∣ + 2 ln ∣ x - 2 ∣ + ln ∣ x ∣ + C

\int \frac{1}{2 x ^{2} + 8 x + 20} d x

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

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