What is the integral of (x^3-x^2)/(x^2-7x+10) ?

The integral of (x^3-x^2)/(x^2-7x+10) is

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integral of (x^3-x^2)/(x^2-7x+10)

\int \frac{x ^{3} - x ^{2}}{x ^{2} - 7 x + 10} d x

\frac{x ^{2}}{2} + 6 x - \frac{8}{3} ln ∣ x - 2 ∣ + \frac{14}{3} ln ∣ 2 x - 4 ∣ - \frac{35}{3} ln ∣ 2 x - 10 ∣ + \frac{125}{3} ln ∣ x - 5 ∣ - \frac{10}{3} (ln \frac{2}{3} x - \frac{4}{3} - ln \frac{2}{3} x - \frac{10}{3}) + C

Solution steps

\int \frac{x ^{3} - x ^{2}}{x ^{2} - 7 x + 10} d x

Expand

\frac{x ^{3} - x ^{2}}{x ^{2} - 7 x + 10} : \frac{x ^{3}}{x ^{2} - 7 x + 10} - \frac{x ^{2}}{x ^{2} - 7 x + 10}

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 ^{3}}{x ^{2} - 7 x + 10} d x - \int \frac{x ^{2}}{x ^{2} - 7 x + 10} d x

\int \frac{x ^{3}}{x ^{2} - 7 x + 10} d x = \frac{x ^{2}}{2} + 7 x - \frac{8}{3} ln ∣ x - 2 ∣ + \frac{125}{3} ln ∣ x - 5 ∣

\int \frac{x ^{2}}{x ^{2} - 7 x + 10} d x = 4 (7 (- \frac{1}{6} ln ∣ 2 x - 4 ∣ + \frac{5}{12} ln ∣ 2 x - 10 ∣) + \frac{5}{6} (ln - \frac{4}{3} + \frac{2}{3} x - ln - \frac{10}{3} + \frac{2}{3} x)) + x

= \frac{x ^{2}}{2} + 7 x - \frac{8}{3} ln ∣ x - 2 ∣ + \frac{125}{3} ln ∣ x - 5 ∣ - (4 (7 (- \frac{1}{6} ln ∣ 2 x - 4 ∣ + \frac{5}{12} ln ∣ 2 x - 10 ∣) + \frac{5}{6} (ln - \frac{4}{3} + \frac{2}{3} x - ln - \frac{10}{3} + \frac{2}{3} x)) + x)

Simplify

\frac{x ^{2}}{2} + 7 x - \frac{8}{3} ln ∣ x - 2 ∣ + \frac{125}{3} ln ∣ x - 5 ∣ - (4 (7 (- \frac{1}{6} ln ∣ 2 x - 4 ∣ + \frac{5}{12} ln ∣ 2 x - 10 ∣) + \frac{5}{6} (ln - \frac{4}{3} + \frac{2}{3} x - ln - \frac{10}{3} + \frac{2}{3} x)) + x) : \frac{x ^{2}}{2} + 6 x - \frac{8}{3} ln ∣ x - 2 ∣ + \frac{14}{3} ln ∣ 2 x - 4 ∣ - \frac{35}{3} ln ∣ 2 x - 10 ∣ + \frac{125}{3} ln ∣ x - 5 ∣ - \frac{10}{3} (ln \frac{2}{3} x - \frac{4}{3} - ln \frac{2}{3} x - \frac{10}{3})

= \frac{x ^{2}}{2} + 6 x - \frac{8}{3} ln ∣ x - 2 ∣ + \frac{14}{3} ln ∣ 2 x - 4 ∣ - \frac{35}{3} ln ∣ 2 x - 10 ∣ + \frac{125}{3} ln ∣ x - 5 ∣ - \frac{10}{3} (ln \frac{2}{3} x - \frac{4}{3} - ln \frac{2}{3} x - \frac{10}{3})

Add a constant to the solution

= \frac{x ^{2}}{2} + 6 x - \frac{8}{3} ln ∣ x - 2 ∣ + \frac{14}{3} ln ∣ 2 x - 4 ∣ - \frac{35}{3} ln ∣ 2 x - 10 ∣ + \frac{125}{3} ln ∣ x - 5 ∣ - \frac{10}{3} (ln \frac{2}{3} x - \frac{4}{3} - ln \frac{2}{3} x - \frac{10}{3}) + C

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What is the integral of (x^3-x^2)/(x^2-7x+10) ?
The integral of (x^3-x^2)/(x^2-7x+10) is