What is the integral of (10x^2-2x-6)/(x^3-x^2-2x) ?

The integral of (10x^2-2x-6)/(x^3-x^2-2x) is

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integral of (10x^2-2x-6)/(x^3-x^2-2x)

\int \frac{10 x ^{2} - 2 x - 6}{x ^{3} - x ^{2} - 2 x} d x

\frac{2}{3} ln \frac{2}{3} x + \frac{2}{3} - \frac{2}{3} ln \frac{2}{3} x - \frac{4}{3} + \frac{4}{3} ln ∣ x + 1 ∣ + \frac{17}{3} ln ∣ x - 2 ∣ + 3 ln ∣ x ∣ + C

Solution steps

\int \frac{10 x ^{2} - 2 x - 6}{x ^{3} - x ^{2} - 2 x} d x

Expand

\frac{10 x ^{2} - 2 x - 6}{x ^{3} - x ^{2} - 2 x} : \frac{10 x ^{2}}{x ^{3} - x ^{2} - 2 x} - \frac{2 x}{x ^{3} - x ^{2} - 2 x} - \frac{6}{x ^{3} - x ^{2} - 2 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 \frac{10 x ^{2}}{x ^{3} - x ^{2} - 2 x} d x - \int \frac{2 x}{x ^{3} - x ^{2} - 2 x} d x - \int \frac{6}{x ^{3} - x ^{2} - 2 x} d x

\int \frac{10 x ^{2}}{x ^{3} - x ^{2} - 2 x} d x = 10 (\frac{1}{3} ln ∣ x + 1 ∣ + \frac{2}{3} ln ∣ x - 2 ∣)

\int \frac{2 x}{x ^{3} - x ^{2} - 2 x} d x = - \frac{2}{3} (ln \frac{2}{3} x + \frac{2}{3} - ln \frac{2}{3} x - \frac{4}{3})

\int \frac{6}{x ^{3} - x ^{2} - 2 x} d x = - 3 ln ∣ x ∣ + 2 ln ∣ x + 1 ∣ + ln ∣ x - 2 ∣

= 10 (\frac{1}{3} ln ∣ x + 1 ∣ + \frac{2}{3} ln ∣ x - 2 ∣) - (- \frac{2}{3} (ln \frac{2}{3} x + \frac{2}{3} - ln \frac{2}{3} x - \frac{4}{3})) - (- 3 ln ∣ x ∣ + 2 ln ∣ x + 1 ∣ + ln ∣ x - 2 ∣)

Simplify

10 (\frac{1}{3} ln ∣ x + 1 ∣ + \frac{2}{3} ln ∣ x - 2 ∣) - (- \frac{2}{3} (ln \frac{2}{3} x + \frac{2}{3} - ln \frac{2}{3} x - \frac{4}{3})) - (- 3 ln ∣ x ∣ + 2 ln ∣ x + 1 ∣ + ln ∣ x - 2 ∣) : \frac{2}{3} ln \frac{2}{3} x + \frac{2}{3} - \frac{2}{3} ln \frac{2}{3} x - \frac{4}{3} + \frac{4}{3} ln ∣ x + 1 ∣ + \frac{17}{3} ln ∣ x - 2 ∣ + 3 ln ∣ x ∣

= \frac{2}{3} ln \frac{2}{3} x + \frac{2}{3} - \frac{2}{3} ln \frac{2}{3} x - \frac{4}{3} + \frac{4}{3} ln ∣ x + 1 ∣ + \frac{17}{3} ln ∣ x - 2 ∣ + 3 ln ∣ x ∣

Add a constant to the solution

= \frac{2}{3} ln \frac{2}{3} x + \frac{2}{3} - \frac{2}{3} ln \frac{2}{3} x - \frac{4}{3} + \frac{4}{3} ln ∣ x + 1 ∣ + \frac{17}{3} ln ∣ x - 2 ∣ + 3 ln ∣ x ∣ + C

\int \frac{1}{2 gx} d x

\int \frac{( x - 1 ) ^{2}}{x} d x

\int x 3 x + 8 d x

\int 3 - \frac{1}{x ^{4}} + \frac{1}{x ^{2}} d x

\int \frac{e ^{5 x}}{e ^{5 x} + 1} d x

What is the integral of (10x^2-2x-6)/(x^3-x^2-2x) ?
The integral of (10x^2-2x-6)/(x^3-x^2-2x) is