What is the integral of (x^2+3x+8)/(x^3-x^2+3x-3) ?

The integral of (x^2+3x+8)/(x^3-x^2+3x-3) is 3ln|x-1|-ln|(x^2)/3+1|+1/(sqrt(3))arctan(x/(sqrt(3)))+C

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integral of (x^2+3x+8)/(x^3-x^2+3x-3)

\int \frac{x ^{2} + 3 x + 8}{x ^{3} - x ^{2} + 3 x - 3} d x

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

Solution steps

\int \frac{x ^{2} + 3 x + 8}{x ^{3} - x ^{2} + 3 x - 3} d x

Take the partial fraction of

\frac{x ^{2} + 3 x + 8}{x ^{3} - x ^{2} + 3 x - 3} : \frac{3}{x - 1} + \frac{- 2 x + 1}{x ^{2} + 3}

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

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

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

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

Add a constant to the solution

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

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What is the integral of (x^2+3x+8)/(x^3-x^2+3x-3) ?
The integral of (x^2+3x+8)/(x^3-x^2+3x-3) is 3ln|x-1|-ln|(x^2)/3+1|+1/(sqrt(3))arctan(x/(sqrt(3)))+C