What is the integral of (4x^2-4x+24)/(x^3+3x) ?

The integral of (4x^2-4x+24)/(x^3+3x) is 8ln|x|-2ln|x^2+3|-4/(sqrt(3))arctan(x/(sqrt(3)))+C

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integral of (4x^2-4x+24)/(x^3+3x)

\int \frac{4 x ^{2} - 4 x + 24}{x ^{3} + 3 x} d x

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

Solution steps

\int \frac{4 x ^{2} - 4 x + 24}{x ^{3} + 3 x} d x

Take the partial fraction of

\frac{4 x ^{2} - 4 x + 24}{x ^{3} + 3 x} : \frac{8}{x} + \frac{- 4 x - 4}{x ^{2} + 3}

= \int \frac{8}{x} + \frac{- 4 x - 4}{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{8}{x} d x + \int \frac{- 4 x - 4}{x ^{2} + 3} d x

\int \frac{8}{x} d x = 8 ln ∣ x ∣

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

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

Add a constant to the solution

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

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What is the integral of (4x^2-4x+24)/(x^3+3x) ?
The integral of (4x^2-4x+24)/(x^3+3x) is 8ln|x|-2ln|x^2+3|-4/(sqrt(3))arctan(x/(sqrt(3)))+C