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

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

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

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

- ln ∣ x - 1 ∣ + \frac{1}{6} (- ln \frac{4 x ^{2} + 4 x}{3} + \frac{4}{3} + 23 arctan (\frac{2 x + 1}{3})) - \frac{2}{3} (- ln 4 x^{2} + 4 x + 4 - 23 arctan (\frac{2}{3} (x - 1) + 3)) + C

Solution steps

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

Expand

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

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

\int \frac{x}{( x - 1 ) ( x ^{2} + x + 1 )} d x = \frac{1}{3} ln ∣ x - 1 ∣ + \frac{1}{6} (- ln \frac{4}{3} + \frac{4 x ^{2} + 4 x}{3} + 23 arctan (\frac{2 x + 1}{3}))

\int \frac{4}{( x - 1 ) ( x ^{2} + x + 1 )} d x = 4 (\frac{1}{3} ln ∣ x - 1 ∣ + \frac{1}{6} (- ln 4 x^{2} + 4 x + 4 - 23 arctan (\frac{2}{3} (x - 1) + 3)))

= \frac{1}{3} ln ∣ x - 1 ∣ + \frac{1}{6} (- ln \frac{4}{3} + \frac{4 x ^{2} + 4 x}{3} + 23 arctan (\frac{2 x + 1}{3})) - 4 (\frac{1}{3} ln ∣ x - 1 ∣ + \frac{1}{6} (- ln 4 x^{2} + 4 x + 4 - 23 arctan (\frac{2}{3} (x - 1) + 3)))

Simplify

\frac{1}{3} ln ∣ x - 1 ∣ + \frac{1}{6} (- ln \frac{4}{3} + \frac{4 x ^{2} + 4 x}{3} + 23 arctan (\frac{2 x + 1}{3})) - 4 (\frac{1}{3} ln ∣ x - 1 ∣ + \frac{1}{6} (- ln 4 x^{2} + 4 x + 4 - 23 arctan (\frac{2}{3} (x - 1) + 3))) : - ln ∣ x - 1 ∣ + \frac{1}{6} (- ln \frac{4 x ^{2} + 4 x}{3} + \frac{4}{3} + 23 arctan (\frac{2 x + 1}{3})) - \frac{2}{3} (- ln 4 x^{2} + 4 x + 4 - 23 arctan (\frac{2}{3} (x - 1) + 3))

= - ln ∣ x - 1 ∣ + \frac{1}{6} (- ln \frac{4 x ^{2} + 4 x}{3} + \frac{4}{3} + 23 arctan (\frac{2 x + 1}{3})) - \frac{2}{3} (- ln 4 x^{2} + 4 x + 4 - 23 arctan (\frac{2}{3} (x - 1) + 3))

Add a constant to the solution

= - ln ∣ x - 1 ∣ + \frac{1}{6} (- ln \frac{4 x ^{2} + 4 x}{3} + \frac{4}{3} + 23 arctan (\frac{2 x + 1}{3})) - \frac{2}{3} (- ln 4 x^{2} + 4 x + 4 - 23 arctan (\frac{2}{3} (x - 1) + 3)) + C

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