What is the integral of (4x^2+9x-1)/(x^3+2x^2-x-2) ?

The integral of (4x^2+9x-1)/(x^3+2x^2-x-2) is -ln|x+2|+3ln|x+1|+2ln|x-1|+C

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integral of (4x^2+9x-1)/(x^3+2x^2-x-2)

\int \frac{4 x ^{2} + 9 x - 1}{x ^{3} + 2 x ^{2} - x - 2} d x

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

Solution steps

\int \frac{4 x ^{2} + 9 x - 1}{x ^{3} + 2 x ^{2} - x - 2} d x

Take the partial fraction of

\frac{4 x ^{2} + 9 x - 1}{x ^{3} + 2 x ^{2} - x - 2} : - \frac{1}{x + 2} + \frac{3}{x + 1} + \frac{2}{x - 1}

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

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

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

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

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

Add a constant to the solution

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

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What is the integral of (4x^2+9x-1)/(x^3+2x^2-x-2) ?
The integral of (4x^2+9x-1)/(x^3+2x^2-x-2) is -ln|x+2|+3ln|x+1|+2ln|x-1|+C