What is the integral of (-x^2+3x+20)/(3x^3+12x) ?

The integral of (-x^2+3x+20)/(3x^3+12x) is 5/3 ln|x|-ln|(x^2)/4+1|+1/2 arctan(x/2)+C

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integral of (-x^2+3x+20)/(3x^3+12x)

\int \frac{- x ^{2} + 3 x + 20}{3 x ^{3} + 12 x} d x

\frac{5}{3} ln ∣ x ∣ - ln \frac{x ^{2}}{4} + 1 + \frac{1}{2} arctan (\frac{x}{2}) + C

Solution steps

\int \frac{- x ^{2} + 3 x + 20}{3 x ^{3} + 12 x} d x

Take the partial fraction of

\frac{- x ^{2} + 3 x + 20}{3 x ^{3} + 12 x} : \frac{5}{3 x} + \frac{- 2 x + 1}{x ^{2} + 4}

= \int \frac{5}{3 x} + \frac{- 2 x + 1}{x ^{2} + 4} 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{5}{3 x} d x + \int \frac{- 2 x + 1}{x ^{2} + 4} d x

\int \frac{5}{3 x} d x = \frac{5}{3} ln ∣ x ∣

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

= \frac{5}{3} ln ∣ x ∣ - ln \frac{x ^{2}}{4} + 1 + \frac{1}{2} arctan (\frac{x}{2})

Add a constant to the solution

= \frac{5}{3} ln ∣ x ∣ - ln \frac{x ^{2}}{4} + 1 + \frac{1}{2} arctan (\frac{x}{2}) + C

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What is the integral of (-x^2+3x+20)/(3x^3+12x) ?
The integral of (-x^2+3x+20)/(3x^3+12x) is 5/3 ln|x|-ln|(x^2)/4+1|+1/2 arctan(x/2)+C