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

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

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

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

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

Solution steps

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

Expand

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

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

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

Simplify

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

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

Add a constant to the solution

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

\int \frac{5 x}{x ^{2} + 2} d x

\int \frac{6 x}{1 - x ^{2}} d x

\int x^{15} ln (3 x^{8}) d x

e^{- t^{2}}

\int sec^{11} (θ) tan (θ) d θ

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