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

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

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

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

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

Solution steps

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

Take the partial fraction of

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

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

\int \frac{2}{x} d x = 2 ln ∣ x ∣

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

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

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

Add a constant to the solution

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

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