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

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

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

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

\frac{1}{2} ln ∣ x + 1 ∣ + \frac{5}{2} ln ∣ x - 1 ∣ - 3 ln ∣ x ∣ + C

Solution steps

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

Expand

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

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

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

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

Simplify

- ln ∣ x + 1 ∣ + ln ∣ x - 1 ∣ + 3 (- ln ∣ x ∣ + \frac{1}{2} ln ∣ x + 1 ∣ + \frac{1}{2} ln ∣ x - 1 ∣) : \frac{1}{2} ln ∣ x + 1 ∣ + \frac{5}{2} ln ∣ x - 1 ∣ - 3 ln ∣ x ∣

= \frac{1}{2} ln ∣ x + 1 ∣ + \frac{5}{2} ln ∣ x - 1 ∣ - 3 ln ∣ x ∣

Add a constant to the solution

= \frac{1}{2} ln ∣ x + 1 ∣ + \frac{5}{2} ln ∣ x - 1 ∣ - 3 ln ∣ x ∣ + C

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