What is the integral of (3x^2-3x+18)/(x^3+3x) ?

The integral of (3x^2-3x+18)/(x^3+3x) is 6ln|x|-3/2 ln|x^2+3|-sqrt(3)arctan(x/(sqrt(3)))+C

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integral of (3x^2-3x+18)/(x^3+3x)

\int \frac{3 x ^{2} - 3 x + 18}{x ^{3} + 3 x} d x

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

Solution steps

\int \frac{3 x ^{2} - 3 x + 18}{x ^{3} + 3 x} d x

Take the partial fraction of

\frac{3 x ^{2} - 3 x + 18}{x ^{3} + 3 x} : \frac{6}{x} + \frac{- 3 x - 3}{x ^{2} + 3}

= \int \frac{6}{x} + \frac{- 3 x - 3}{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{6}{x} d x + \int \frac{- 3 x - 3}{x ^{2} + 3} d x

\int \frac{6}{x} d x = 6 ln ∣ x ∣

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

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

Add a constant to the solution

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

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What is the integral of (3x^2-3x+18)/(x^3+3x) ?
The integral of (3x^2-3x+18)/(x^3+3x) is 6ln|x|-3/2 ln|x^2+3|-sqrt(3)arctan(x/(sqrt(3)))+C