What is the integral of (28x^3-56x^2+9)/(x^2-2x) ?

The integral of (28x^3-56x^2+9)/(x^2-2x) is 14x^2-9(1/2 ln|x|-1/2 ln|x-2|)+C

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integral of (28x^3-56x^2+9)/(x^2-2x)

\int \frac{28 x ^{3} - 56 x ^{2} + 9}{x ^{2} - 2 x} d x

14 x^{2} - 9 (\frac{1}{2} ln ∣ x ∣ - \frac{1}{2} ln ∣ x - 2 ∣) + C

Solution steps

\int \frac{28 x ^{3} - 56 x ^{2} + 9}{x ^{2} - 2 x} d x

Long division

\frac{28 x ^{3} - 56 x ^{2} + 9}{x ^{2} - 2 x} : 28 x + \frac{9}{x ^{2} - 2 x}

= \int 28 x + \frac{9}{x ^{2} - 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 28 x d x + \int \frac{9}{x ^{2} - 2 x} d x

\int 28 x d x = 14 x^{2}

\int \frac{9}{x ^{2} - 2 x} d x = - 9 (\frac{1}{2} ln ∣ x ∣ - \frac{1}{2} ln ∣ x - 2 ∣)

= 14 x^{2} - 9 (\frac{1}{2} ln ∣ x ∣ - \frac{1}{2} ln ∣ x - 2 ∣)

Add a constant to the solution

= 14 x^{2} - 9 (\frac{1}{2} ln ∣ x ∣ - \frac{1}{2} ln ∣ x - 2 ∣) + C

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What is the integral of (28x^3-56x^2+9)/(x^2-2x) ?
The integral of (28x^3-56x^2+9)/(x^2-2x) is 14x^2-9(1/2 ln|x|-1/2 ln|x-2|)+C