What is the integral of (5x-14)/(x^3-x^2-4x+4) ?

The integral of (5x-14)/(x^3-x^2-4x+4) is 3ln|x-1|-2ln|x+2|-ln|x-2|+C

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integral of (5x-14)/(x^3-x^2-4x+4)

\int \frac{5 x - 14}{x ^{3} - x ^{2} - 4 x + 4} d x

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

Solution steps

\int \frac{5 x - 14}{x ^{3} - x ^{2} - 4 x + 4} d x

Take the partial fraction of

\frac{5 x - 14}{x ^{3} - x ^{2} - 4 x + 4} : \frac{3}{x - 1} - \frac{2}{x + 2} - \frac{1}{x - 2}

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

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

\int \frac{2}{x + 2} d x = 2 ln ∣ x + 2 ∣

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

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

Add a constant to the solution

= 3 ln ∣ x - 1 ∣ - 2 ln ∣ x + 2 ∣ - ln ∣ x - 2 ∣ + C

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What is the integral of (5x-14)/(x^3-x^2-4x+4) ?
The integral of (5x-14)/(x^3-x^2-4x+4) is 3ln|x-1|-2ln|x+2|-ln|x-2|+C