What is the integral of (3x^2+7x-2)/(x^3-x^2-2x) ?

The integral of (3x^2+7x-2)/(x^3-x^2-2x) is ln|x|-2ln|x+1|+4ln|x-2|+C

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integral of (3x^2+7x-2)/(x^3-x^2-2x)

\int \frac{3 x ^{2} + 7 x - 2}{x ^{3} - x ^{2} - 2 x} d x

ln ∣ x ∣ - 2 ln ∣ x + 1 ∣ + 4 ln ∣ x - 2 ∣ + C

Solution steps

\int \frac{3 x ^{2} + 7 x - 2}{x ^{3} - x ^{2} - 2 x} d x

Take the partial fraction of

\frac{3 x ^{2} + 7 x - 2}{x ^{3} - x ^{2} - 2 x} : \frac{1}{x} - \frac{2}{x + 1} + \frac{4}{x - 2}

= \int \frac{1}{x} - \frac{2}{x + 1} + \frac{4}{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{1}{x} d x - \int \frac{2}{x + 1} d x + \int \frac{4}{x - 2} d x

\int \frac{1}{x} d x = ln ∣ x ∣

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

\int \frac{4}{x - 2} d x = 4 ln ∣ x - 2 ∣

= ln ∣ x ∣ - 2 ln ∣ x + 1 ∣ + 4 ln ∣ x - 2 ∣

Add a constant to the solution

= ln ∣ x ∣ - 2 ln ∣ x + 1 ∣ + 4 ln ∣ x - 2 ∣ + C

What is the integral of (3x^2+7x-2)/(x^3-x^2-2x) ?
The integral of (3x^2+7x-2)/(x^3-x^2-2x) is ln|x|-2ln|x+1|+4ln|x-2|+C