What is the integral of (26x^3-52x^2+8)/(x^2-2x) ?

The integral of (26x^3-52x^2+8)/(x^2-2x) is 13x^2-8(1/2 ln|x|-1/2 ln|x-2|)+C

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integral of (26x^3-52x^2+8)/(x^2-2x)

\int \frac{26 x ^{3} - 52 x ^{2} + 8}{x ^{2} - 2 x} d x

13 x^{2} - 8 (\frac{1}{2} ln ∣ x ∣ - \frac{1}{2} ln ∣ x - 2 ∣) + C

Solution steps

\int \frac{26 x ^{3} - 52 x ^{2} + 8}{x ^{2} - 2 x} d x

Long division

\frac{26 x ^{3} - 52 x ^{2} + 8}{x ^{2} - 2 x} : 26 x + \frac{8}{x ^{2} - 2 x}

= \int 26 x + \frac{8}{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 26 x d x + \int \frac{8}{x ^{2} - 2 x} d x

\int 26 x d x = 13 x^{2}

\int \frac{8}{x ^{2} - 2 x} d x = - 8 (\frac{1}{2} ln ∣ x ∣ - \frac{1}{2} ln ∣ x - 2 ∣)

= 13 x^{2} - 8 (\frac{1}{2} ln ∣ x ∣ - \frac{1}{2} ln ∣ x - 2 ∣)

Add a constant to the solution

= 13 x^{2} - 8 (\frac{1}{2} ln ∣ x ∣ - \frac{1}{2} ln ∣ x - 2 ∣) + C

\int cot (16 x) d x

x \to 0 - lim (\frac{5}{x ^{3}})

x^{18}

\frac{d}{d x} (tan (2 x) cot (2 x))

35

What is the integral of (26x^3-52x^2+8)/(x^2-2x) ?
The integral of (26x^3-52x^2+8)/(x^2-2x) is 13x^2-8(1/2 ln|x|-1/2 ln|x-2|)+C