What is the integral of (x+8)/(x^3+4x) ?

The integral of (x+8)/(x^3+4x) is 2ln|x|-ln|(x^2)/4+1|+1/2 arctan(x/2)+C

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integral of (x+8)/(x^3+4x)

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

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

Solution steps

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

Take the partial fraction of

\frac{x + 8}{x ^{3} + 4 x} : \frac{2}{x} + \frac{- 2 x + 1}{x ^{2} + 4}

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

\int \frac{2}{x} d x = 2 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})

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

Add a constant to the solution

= 2 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+8)/(x^3+4x) ?
The integral of (x+8)/(x^3+4x) is 2ln|x|-ln|(x^2)/4+1|+1/2 arctan(x/2)+C