What is the integral of (x^2-5x+1)/(x^2-5x+4) ?

The integral of (x^2-5x+1)/(x^2-5x+4) is

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integral of (x^2-5x+1)/(x^2-5x+4)

\int \frac{x ^{2} - 5 x + 1}{x ^{2} - 5 x + 4} d x

4 (5 (- \frac{1}{12} ln ∣ 2 x - 2 ∣ + \frac{1}{3} ln ∣ 2 x - 8 ∣) + \frac{1}{3} (ln - \frac{2}{3} + \frac{2}{3} x - ln - \frac{8}{3} + \frac{2}{3} x)) + x - 10 (\frac{1}{4} ln 4 x^{2} - 20 x + 16 - \frac{5}{12} (ln - \frac{2}{3} + \frac{2}{3} x - ln - \frac{8}{3} + \frac{2}{3} x)) - \frac{1}{3} ln ∣ x - 1 ∣ + \frac{1}{3} ln ∣ x - 4 ∣ + C

Solution steps

\int \frac{x ^{2} - 5 x + 1}{x ^{2} - 5 x + 4} d x

Expand

\frac{x ^{2} - 5 x + 1}{x ^{2} - 5 x + 4} : \frac{x ^{2}}{x ^{2} - 5 x + 4} - \frac{5 x}{x ^{2} - 5 x + 4} + \frac{1}{x ^{2} - 5 x + 4}

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{x ^{2}}{x ^{2} - 5 x + 4} d x - \int \frac{5 x}{x ^{2} - 5 x + 4} d x + \int \frac{1}{x ^{2} - 5 x + 4} d x

\int \frac{x ^{2}}{x ^{2} - 5 x + 4} d x = 4 (5 (- \frac{1}{12} ln ∣ 2 x - 2 ∣ + \frac{1}{3} ln ∣ 2 x - 8 ∣) + \frac{1}{3} (ln - \frac{2}{3} + \frac{2}{3} x - ln - \frac{8}{3} + \frac{2}{3} x)) + x

\int \frac{5 x}{x ^{2} - 5 x + 4} d x = 10 (\frac{1}{4} ln 4 x^{2} - 20 x + 16 - \frac{5}{12} (ln - \frac{2}{3} + \frac{2}{3} x - ln - \frac{8}{3} + \frac{2}{3} x))

\int \frac{1}{x ^{2} - 5 x + 4} d x = - \frac{1}{3} ln ∣ x - 1 ∣ + \frac{1}{3} ln ∣ x - 4 ∣

= 4 (5 (- \frac{1}{12} ln ∣ 2 x - 2 ∣ + \frac{1}{3} ln ∣ 2 x - 8 ∣) + \frac{1}{3} (ln - \frac{2}{3} + \frac{2}{3} x - ln - \frac{8}{3} + \frac{2}{3} x)) + x - 10 (\frac{1}{4} ln 4 x^{2} - 20 x + 16 - \frac{5}{12} (ln - \frac{2}{3} + \frac{2}{3} x - ln - \frac{8}{3} + \frac{2}{3} x)) - \frac{1}{3} ln ∣ x - 1 ∣ + \frac{1}{3} ln ∣ x - 4 ∣

Add a constant to the solution

= 4 (5 (- \frac{1}{12} ln ∣ 2 x - 2 ∣ + \frac{1}{3} ln ∣ 2 x - 8 ∣) + \frac{1}{3} (ln - \frac{2}{3} + \frac{2}{3} x - ln - \frac{8}{3} + \frac{2}{3} x)) + x - 10 (\frac{1}{4} ln 4 x^{2} - 20 x + 16 - \frac{5}{12} (ln - \frac{2}{3} + \frac{2}{3} x - ln - \frac{8}{3} + \frac{2}{3} x)) - \frac{1}{3} ln ∣ x - 1 ∣ + \frac{1}{3} ln ∣ x - 4 ∣ + C

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What is the integral of (x^2-5x+1)/(x^2-5x+4) ?
The integral of (x^2-5x+1)/(x^2-5x+4) is