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

The integral of (x^3)/(x^2-4x+4) is (x^2)/2+4x-(12x-16)/(x-2)+12ln|x-2|+C

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

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

\frac{x ^{2}}{2} + 4 x - \frac{12 x - 16}{x - 2} + 12 ln ∣ x - 2 ∣ + C

Solution steps

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

Long division

\frac{x ^{3}}{x ^{2} - 4 x + 4} : x + 4 + \frac{12 x - 16}{x ^{2} - 4 x + 4}

= \int x + 4 + \frac{12 x - 16}{x ^{2} - 4 x + 4} 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 4 d x + \int \frac{12 x - 16}{x ^{2} - 4 x + 4} d x

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

\int 4 d x = 4 x

\int \frac{12 x - 16}{x ^{2} - 4 x + 4} d x = - \frac{12 x - 16}{x - 2} + 12 ln ∣ x - 2 ∣

= \frac{x ^{2}}{2} + 4 x - \frac{12 x - 16}{x - 2} + 12 ln ∣ x - 2 ∣

Add a constant to the solution

= \frac{x ^{2}}{2} + 4 x - \frac{12 x - 16}{x - 2} + 12 ln ∣ x - 2 ∣ + C

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What is the integral of (x^3)/(x^2-4x+4) ?
The integral of (x^3)/(x^2-4x+4) is (x^2)/2+4x-(12x-16)/(x-2)+12ln|x-2|+C