What is the integral of (24x^3-48x^2+5)/(x^2-2x) ?

The integral of (24x^3-48x^2+5)/(x^2-2x) is 12x^2-5(1/2 ln|x|-1/2 ln|x-2|)+C

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integral of (24x^3-48x^2+5)/(x^2-2x)

\int \frac{24 x ^{3} - 48 x ^{2} + 5}{x ^{2} - 2 x} d x

12 x^{2} - 5 (\frac{1}{2} ln ∣ x ∣ - \frac{1}{2} ln ∣ x - 2 ∣) + C

Solution steps

\int \frac{24 x ^{3} - 48 x ^{2} + 5}{x ^{2} - 2 x} d x

Long division

\frac{24 x ^{3} - 48 x ^{2} + 5}{x ^{2} - 2 x} : 24 x + \frac{5}{x ^{2} - 2 x}

= \int 24 x + \frac{5}{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{5}{x ^{2} - 2 x} d x

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

\int \frac{5}{x ^{2} - 2 x} d x = - 5 (\frac{1}{2} ln ∣ x ∣ - \frac{1}{2} ln ∣ x - 2 ∣)

= 12 x^{2} - 5 (\frac{1}{2} ln ∣ x ∣ - \frac{1}{2} ln ∣ x - 2 ∣)

Add a constant to the solution

= 12 x^{2} - 5 (\frac{1}{2} ln ∣ x ∣ - \frac{1}{2} ln ∣ x - 2 ∣) + C

\frac{1}{4 x + 3}

x - x^{2} - x^{2} + 6 x, 0, 3.5

y = x^{2}, y = 0, x = 1

\int \frac{w ^{2} - 25}{w ^{3}} d w

0.03 \cdot (1 - cos (2 t))

What is the integral of (24x^3-48x^2+5)/(x^2-2x) ?
The integral of (24x^3-48x^2+5)/(x^2-2x) is 12x^2-5(1/2 ln|x|-1/2 ln|x-2|)+C