What is the integral of (x^3+x+2)/(x^2+2x-8) ?

The integral of (x^3+x+2)/(x^2+2x-8) is (x^2)/2-2x+2ln|x-2|+11ln|x+4|+C

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integral of (x^3+x+2)/(x^2+2x-8)

\int \frac{x ^{3} + x + 2}{x ^{2} + 2 x - 8} d x

\frac{x ^{2}}{2} - 2 x + 2 ln ∣ x - 2 ∣ + 11 ln ∣ x + 4 ∣ + C

Solution steps

\int \frac{x ^{3} + x + 2}{x ^{2} + 2 x - 8} d x

Long division

\frac{x ^{3} + x + 2}{x ^{2} + 2 x - 8} : x - 2 + \frac{13 x - 14}{x ^{2} + 2 x - 8}

= \int x - 2 + \frac{13 x - 14}{x ^{2} + 2 x - 8} 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 2 d x + \int \frac{13 x - 14}{x ^{2} + 2 x - 8} d x

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

\int 2 d x = 2 x

\int \frac{13 x - 14}{x ^{2} + 2 x - 8} d x = 2 ln ∣ x - 2 ∣ + 11 ln ∣ x + 4 ∣

= \frac{x ^{2}}{2} - 2 x + 2 ln ∣ x - 2 ∣ + 11 ln ∣ x + 4 ∣

Add a constant to the solution

= \frac{x ^{2}}{2} - 2 x + 2 ln ∣ x - 2 ∣ + 11 ln ∣ x + 4 ∣ + C

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What is the integral of (x^3+x+2)/(x^2+2x-8) ?
The integral of (x^3+x+2)/(x^2+2x-8) is (x^2)/2-2x+2ln|x-2|+11ln|x+4|+C