What is the integral of (5x^2-9)/(x^3-9x) ?

The integral of (5x^2-9)/(x^3-9x) is 5/2 ln(|x+3||x-3|)+ln|x|-1/2 ln|x+3|-1/2 ln|x-3|+C

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integral of (5x^2-9)/(x^3-9x)

\int \frac{5 x ^{2} - 9}{x ^{3} - 9 x} d x

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

Solution steps

\int \frac{5 x ^{2} - 9}{x ^{3} - 9 x} d x

Expand

\frac{5 x ^{2} - 9}{x ^{3} - 9 x} : \frac{5 x ^{2}}{x ^{3} - 9 x} - \frac{9}{x ^{3} - 9 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{5 x ^{2}}{x ^{3} - 9 x} d x - \int \frac{9}{x ^{3} - 9 x} d x

\int \frac{5 x ^{2}}{x ^{3} - 9 x} d x = \frac{5}{2} ln (∣ x + 3 ∣ ∣ x - 3 ∣)

\int \frac{9}{x ^{3} - 9 x} d x = - ln ∣ x ∣ + \frac{1}{2} ln ∣ x + 3 ∣ + \frac{1}{2} ln ∣ x - 3 ∣

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

Simplify

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

Add a constant to the solution

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

What is the integral of (5x^2-9)/(x^3-9x) ?
The integral of (5x^2-9)/(x^3-9x) is 5/2 ln(|x+3||x-3|)+ln|x|-1/2 ln|x+3|-1/2 ln|x-3|+C