What is the integral of (24x^3-48x^2+7)/(x^2-2x) ?

The integral of (24x^3-48x^2+7)/(x^2-2x) is 12x^2-7(1/2 ln|x|-1/2 ln|x-2|)+C

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integral of (24x^3-48x^2+7)/(x^2-2x)

\int \frac{24 x ^{3} - 48 x ^{2} + 7}{x ^{2} - 2 x} d x

12 x^{2} - 7 (\frac{1}{2} ln ∣ x ∣ - \frac{1}{2} ln ∣ x - 2 ∣) + C

Solution steps

\int \frac{24 x ^{3} - 48 x ^{2} + 7}{x ^{2} - 2 x} d x

Long division

\frac{24 x ^{3} - 48 x ^{2} + 7}{x ^{2} - 2 x} : 24 x + \frac{7}{x ^{2} - 2 x}

= \int 24 x + \frac{7}{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 24 x d x + \int \frac{7}{x ^{2} - 2 x} d x

\int 24 x d x = 12 x^{2}

\int \frac{7}{x ^{2} - 2 x} d x = - 7 (\frac{1}{2} ln ∣ x ∣ - \frac{1}{2} ln ∣ x - 2 ∣)

= 12 x^{2} - 7 (\frac{1}{2} ln ∣ x ∣ - \frac{1}{2} ln ∣ x - 2 ∣)

Add a constant to the solution

= 12 x^{2} - 7 (\frac{1}{2} ln ∣ x ∣ - \frac{1}{2} ln ∣ x - 2 ∣) + C

(2 x ln (x) + x^{2} - 1)^{^{'}}

\frac{\partial}{\partial x} (u (x))

\int (y^{2} e^{x y}) d x

\frac{\partial}{\partial x} (\frac{x - 2 y}{x + 2 y})

What is the integral of (24x^3-48x^2+7)/(x^2-2x) ?
The integral of (24x^3-48x^2+7)/(x^2-2x) is 12x^2-7(1/2 ln|x|-1/2 ln|x-2|)+C