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

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

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

\int \frac{2 x + 3}{x ( x - 3 ) ( x + 3 )} d x

\frac{1}{2} ln ∣ x - 3 ∣ - \frac{1}{6} ln ∣ x + 3 ∣ - \frac{1}{3} ln ∣ x ∣ + C

Solution steps

\int \frac{2 x + 3}{x ( x - 3 ) ( x + 3 )} d x

Expand

\frac{2 x + 3}{x ( x - 3 ) ( x + 3 )} : \frac{2 x}{x ( x - 3 ) ( x + 3 )} + \frac{3}{x ( x - 3 ) ( x + 3 )}

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 ( x - 3 ) ( x + 3 )} d x + \int \frac{3}{x ( x - 3 ) ( x + 3 )} d x

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

\int \frac{3}{x ( x - 3 ) ( x + 3 )} d x = 3 (\frac{1}{18} ln ∣ x + 3 ∣ - \frac{1}{9} ln ∣ x ∣ + \frac{1}{18} ln ∣ x - 3 ∣)

= 2 (\frac{1}{6} ln ∣ x - 3 ∣ - \frac{1}{6} ln ∣ x + 3 ∣) + 3 (\frac{1}{18} ln ∣ x + 3 ∣ - \frac{1}{9} ln ∣ x ∣ + \frac{1}{18} ln ∣ x - 3 ∣)

Simplify

2 (\frac{1}{6} ln ∣ x - 3 ∣ - \frac{1}{6} ln ∣ x + 3 ∣) + 3 (\frac{1}{18} ln ∣ x + 3 ∣ - \frac{1}{9} ln ∣ x ∣ + \frac{1}{18} ln ∣ x - 3 ∣) : \frac{1}{2} ln ∣ x - 3 ∣ - \frac{1}{6} ln ∣ x + 3 ∣ - \frac{1}{3} ln ∣ x ∣

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

Add a constant to the solution

= \frac{1}{2} ln ∣ x - 3 ∣ - \frac{1}{6} ln ∣ x + 3 ∣ - \frac{1}{3} ln ∣ x ∣ + C

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