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

The integral of (x^3)/(x^2-2x-3) is (x^2)/2+2x+1/4 ln|x+1|+27/4 ln|x-3|+C

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

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

\frac{x ^{2}}{2} + 2 x + \frac{1}{4} ln ∣ x + 1 ∣ + \frac{27}{4} ln ∣ x - 3 ∣ + C

Solution steps

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

Long division

\frac{x ^{3}}{x ^{2} - 2 x - 3} : x + 2 + \frac{7 x + 6}{x ^{2} - 2 x - 3}

= \int x + 2 + \frac{7 x + 6}{x ^{2} - 2 x - 3} 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{7 x + 6}{x ^{2} - 2 x - 3} d x

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

\int 2 d x = 2 x

\int \frac{7 x + 6}{x ^{2} - 2 x - 3} d x = \frac{1}{4} ln ∣ x + 1 ∣ + \frac{27}{4} ln ∣ x - 3 ∣

= \frac{x ^{2}}{2} + 2 x + \frac{1}{4} ln ∣ x + 1 ∣ + \frac{27}{4} ln ∣ x - 3 ∣

Add a constant to the solution

= \frac{x ^{2}}{2} + 2 x + \frac{1}{4} ln ∣ x + 1 ∣ + \frac{27}{4} ln ∣ x - 3 ∣ + C

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What is the integral of (x^3)/(x^2-2x-3) ?
The integral of (x^3)/(x^2-2x-3) is (x^2)/2+2x+1/4 ln|x+1|+27/4 ln|x-3|+C