What is the integral of (12+10x-2x^2)/(x^3-4x) ?

The integral of (12+10x-2x^2)/(x^3-4x) is -3ln|x|-2ln|x+2|+3ln|x-2|+C

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integral of (12+10x-2x^2)/(x^3-4x)

\int \frac{12 + 10 x - 2 x ^{2}}{x ^{3} - 4 x} d x

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

Solution steps

\int \frac{12 + 10 x - 2 x ^{2}}{x ^{3} - 4 x} d x

Take the partial fraction of

\frac{12 + 10 x - 2 x ^{2}}{x ^{3} - 4 x} : - \frac{3}{x} - \frac{2}{x + 2} + \frac{3}{x - 2}

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

\int \frac{3}{x} d x = 3 ln ∣ x ∣

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

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

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

Add a constant to the solution

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

What is the integral of (12+10x-2x^2)/(x^3-4x) ?
The integral of (12+10x-2x^2)/(x^3-4x) is -3ln|x|-2ln|x+2|+3ln|x-2|+C