What is the integral of (x^3-2x-4)/(x^2-2x) ?

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

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integral of (x^3-2x-4)/(x^2-2x)

\int \frac{x ^{3} - 2 x - 4}{x ^{2} - 2 x} d x

\frac{x ^{2}}{2} + 2 x + 2 ln ∣ x ∣ + C

Solution steps

\int \frac{x ^{3} - 2 x - 4}{x ^{2} - 2 x} d x

Simplify

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

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

Expand

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

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

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

\int 2 d x = 2 x

\int \frac{2}{x} d x = 2 ln ∣ x ∣

= \frac{x ^{2}}{2} + 2 x + 2 ln ∣ x ∣

Add a constant to the solution

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

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