What is the integral of (x-4)/(x^3-x^2-x-2) ?

The integral of (x-4)/(x^3-x^2-x-2) is

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integral of (x-4)/(x^3-x^2-x-2)

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

- \frac{2}{7} ln ∣ x - 2 ∣ - \frac{1}{7} ln 4 x^{2} + 4 x + 4 + \frac{8 3}{7} arctan (\frac{1}{3} (2 x + 1)) + \frac{2}{7} ln 4 x^{2} + 4 x + 4 + C

Solution steps

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

Expand

\frac{x - 4}{x ^{3} - x ^{2} - x - 2} : \frac{x}{x ^{3} - x ^{2} - x - 2} - \frac{4}{x ^{3} - x ^{2} - x - 2}

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

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

\int \frac{4}{x ^{3} - x ^{2} - x - 2} d x = 4 (\frac{1}{7} ln ∣ x - 2 ∣ + \frac{1}{7} (- \frac{1}{2} ln 4 x^{2} + 4 x + 4 + \frac{1}{3} arctan (\frac{1}{3} (2 x + 1)) - 23 arctan (\frac{1}{3} (2 x + 1))))

= \frac{2}{7} ln ∣ x - 2 ∣ - \frac{1}{7} ln 4 x^{2} + 4 x + 4 + \frac{4}{7 3} arctan (\frac{1}{3} (2 x + 1)) - 4 (\frac{1}{7} ln ∣ x - 2 ∣ + \frac{1}{7} (- \frac{1}{2} ln 4 x^{2} + 4 x + 4 + \frac{1}{3} arctan (\frac{1}{3} (2 x + 1)) - 23 arctan (\frac{1}{3} (2 x + 1))))

Simplify

\frac{2}{7} ln ∣ x - 2 ∣ - \frac{1}{7} ln 4 x^{2} + 4 x + 4 + \frac{4}{7 3} arctan (\frac{1}{3} (2 x + 1)) - 4 (\frac{1}{7} ln ∣ x - 2 ∣ + \frac{1}{7} (- \frac{1}{2} ln 4 x^{2} + 4 x + 4 + \frac{1}{3} arctan (\frac{1}{3} (2 x + 1)) - 23 arctan (\frac{1}{3} (2 x + 1)))) : - \frac{2}{7} ln ∣ x - 2 ∣ - \frac{1}{7} ln 4 x^{2} + 4 x + 4 + \frac{8 3}{7} arctan (\frac{1}{3} (2 x + 1)) + \frac{2}{7} ln 4 x^{2} + 4 x + 4

= - \frac{2}{7} ln ∣ x - 2 ∣ - \frac{1}{7} ln 4 x^{2} + 4 x + 4 + \frac{8 3}{7} arctan (\frac{1}{3} (2 x + 1)) + \frac{2}{7} ln 4 x^{2} + 4 x + 4

Add a constant to the solution

= - \frac{2}{7} ln ∣ x - 2 ∣ - \frac{1}{7} ln 4 x^{2} + 4 x + 4 + \frac{8 3}{7} arctan (\frac{1}{3} (2 x + 1)) + \frac{2}{7} ln 4 x^{2} + 4 x + 4 + C

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What is the integral of (x-4)/(x^3-x^2-x-2) ?
The integral of (x-4)/(x^3-x^2-x-2) is