What is the integral of (12x^3-24x^2+7)/(x^2-2x) ?

The integral of (12x^3-24x^2+7)/(x^2-2x) is 6x^2-7(1/2 ln|x|-1/2 ln|x-2|)+C

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integral of (12x^3-24x^2+7)/(x^2-2x)

\int \frac{12 x ^{3} - 24 x ^{2} + 7}{x ^{2} - 2 x} d x

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

Solution steps

\int \frac{12 x ^{3} - 24 x ^{2} + 7}{x ^{2} - 2 x} d x

Long division

\frac{12 x ^{3} - 24 x ^{2} + 7}{x ^{2} - 2 x} : 12 x + \frac{7}{x ^{2} - 2 x}

= \int 12 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 12 x d x + \int \frac{7}{x ^{2} - 2 x} d x

\int 12 x d x = 6 x^{2}

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

= 6 x^{2} - 7 (\frac{1}{2} ln ∣ x ∣ - \frac{1}{2} ln ∣ x - 2 ∣)

Add a constant to the solution

= 6 x^{2} - 7 (\frac{1}{2} ln ∣ x ∣ - \frac{1}{2} ln ∣ x - 2 ∣) + C

\frac{s - 3}{s + 7}

x \to - 2 lim (\frac{- 1}{- x ^{2} + 4})

\frac{d y}{d x} = 4 x + \frac{9 x ^{2}}{( 3 x ^{3} + 1 ) ^{\frac{3}{2}}}

\int_{0}^{1} (3 x + 1) cos (x) d x

\frac{d}{d x} (- 9 x^{2} + 4 x + 5)

What is the integral of (12x^3-24x^2+7)/(x^2-2x) ?
The integral of (12x^3-24x^2+7)/(x^2-2x) is 6x^2-7(1/2 ln|x|-1/2 ln|x-2|)+C