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

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

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

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

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

Solution steps

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

Expand

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

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

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

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

Simplify

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

Add a constant to the solution

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

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