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

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

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

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

\frac{x ^{2}}{2} + 2 (\frac{1}{3} ln ∣ x - 1 ∣ + \frac{2}{3} ln ∣ x + 2 ∣) + C

Solution steps

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

Long division

\frac{x ^{3} + x ^{2}}{x ^{2} + x - 2} : x + \frac{2 x}{x ^{2} + x - 2}

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

\int x d x = \frac{x ^{2}}{2}

\int \frac{2 x}{x ^{2} + x - 2} d x = 2 (\frac{1}{3} ln ∣ x - 1 ∣ + \frac{2}{3} ln ∣ x + 2 ∣)

= \frac{x ^{2}}{2} + 2 (\frac{1}{3} ln ∣ x - 1 ∣ + \frac{2}{3} ln ∣ x + 2 ∣)

Add a constant to the solution

= \frac{x ^{2}}{2} + 2 (\frac{1}{3} ln ∣ x - 1 ∣ + \frac{2}{3} ln ∣ x + 2 ∣) + C

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