What is the integral of (3x-1)/(x^3+3x) ?

The integral of (3x-1)/(x^3+3x) is sqrt(3)arctan(x/(sqrt(3)))-1/3 ln|x|+1/6 ln|x^2+3|+C

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integral of (3x-1)/(x^3+3x)

\int \frac{3 x - 1}{x ^{3} + 3 x} d x

3 arctan (\frac{x}{3}) - \frac{1}{3} ln ∣ x ∣ + \frac{1}{6} ln x^{2} + 3 + C

Solution steps

\int \frac{3 x - 1}{x ^{3} + 3 x} d x

Expand

\frac{3 x - 1}{x ^{3} + 3 x} : \frac{3 x}{x ^{3} + 3 x} - \frac{1}{x ^{3} + 3 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{3 x}{x ^{3} + 3 x} d x - \int \frac{1}{x ^{3} + 3 x} d x

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

\int \frac{1}{x ^{3} + 3 x} d x = \frac{1}{3} ln ∣ x ∣ - \frac{1}{6} ln x^{2} + 3

= 3 arctan (\frac{x}{3}) - (\frac{1}{3} ln ∣ x ∣ - \frac{1}{6} ln x^{2} + 3)

Simplify

= 3 arctan (\frac{x}{3}) - \frac{1}{3} ln ∣ x ∣ + \frac{1}{6} ln x^{2} + 3

Add a constant to the solution

= 3 arctan (\frac{x}{3}) - \frac{1}{3} ln ∣ x ∣ + \frac{1}{6} ln x^{2} + 3 + C

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What is the integral of (3x-1)/(x^3+3x) ?
The integral of (3x-1)/(x^3+3x) is sqrt(3)arctan(x/(sqrt(3)))-1/3 ln|x|+1/6 ln|x^2+3|+C