What is the integral of (2x^2-2x+12)/(x^3+3x) ?

The integral of (2x^2-2x+12)/(x^3+3x) is 4ln|x|-ln|x^2+3|-2/(sqrt(3))arctan(x/(sqrt(3)))+C

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integral of (2x^2-2x+12)/(x^3+3x)

\int \frac{2 x ^{2} - 2 x + 12}{x ^{3} + 3 x} d x

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

Solution steps

\int \frac{2 x ^{2} - 2 x + 12}{x ^{3} + 3 x} d x

Take the partial fraction of

\frac{2 x ^{2} - 2 x + 12}{x ^{3} + 3 x} : \frac{4}{x} + \frac{- 2 x - 2}{x ^{2} + 3}

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

\int \frac{4}{x} d x = 4 ln ∣ x ∣

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

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

Add a constant to the solution

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

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What is the integral of (2x^2-2x+12)/(x^3+3x) ?
The integral of (2x^2-2x+12)/(x^3+3x) is 4ln|x|-ln|x^2+3|-2/(sqrt(3))arctan(x/(sqrt(3)))+C