What is the integral of (2x^2-5x-18)/(x^3+x^2-6x) ?

The integral of (2x^2-5x-18)/(x^3+x^2-6x) is 3ln|x|-2ln|x-2|+ln|x+3|+C

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integral of (2x^2-5x-18)/(x^3+x^2-6x)

\int \frac{2 x ^{2} - 5 x - 18}{x ^{3} + x ^{2} - 6 x} d x

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

Solution steps

\int \frac{2 x ^{2} - 5 x - 18}{x ^{3} + x ^{2} - 6 x} d x

Take the partial fraction of

\frac{2 x ^{2} - 5 x - 18}{x ^{3} + x ^{2} - 6 x} : \frac{3}{x} - \frac{2}{x - 2} + \frac{1}{x + 3}

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

\int \frac{3}{x} d x = 3 ln ∣ x ∣

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

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

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

Add a constant to the solution

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

What is the integral of (2x^2-5x-18)/(x^3+x^2-6x) ?
The integral of (2x^2-5x-18)/(x^3+x^2-6x) is 3ln|x|-2ln|x-2|+ln|x+3|+C