What is the integral of (2x^2+5x-1)/(x^3+x^2-2x) ?

The integral of (2x^2+5x-1)/(x^3+x^2-2x) is 1/2 ln|x|+2ln|x-1|-1/2 ln|x+2|+C

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integral of (2x^2+5x-1)/(x^3+x^2-2x)

\int \frac{2 x ^{2} + 5 x - 1}{x ^{3} + x ^{2} - 2 x} d x

\frac{1}{2} ln ∣ x ∣ + 2 ln ∣ x - 1 ∣ - \frac{1}{2} ln ∣ x + 2 ∣ + C

Solution steps

\int \frac{2 x ^{2} + 5 x - 1}{x ^{3} + x ^{2} - 2 x} d x

Take the partial fraction of

\frac{2 x ^{2} + 5 x - 1}{x ^{3} + x ^{2} - 2 x} : \frac{1}{2 x} + \frac{2}{x - 1} - \frac{1}{2 ( x + 2 )}

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

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

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

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

= \frac{1}{2} ln ∣ x ∣ + 2 ln ∣ x - 1 ∣ - \frac{1}{2} ln ∣ x + 2 ∣

Add a constant to the solution

= \frac{1}{2} ln ∣ x ∣ + 2 ln ∣ x - 1 ∣ - \frac{1}{2} ln ∣ x + 2 ∣ + C

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What is the integral of (2x^2+5x-1)/(x^3+x^2-2x) ?
The integral of (2x^2+5x-1)/(x^3+x^2-2x) is 1/2 ln|x|+2ln|x-1|-1/2 ln|x+2|+C