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

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

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integral of (x^2+9)/(x(x^2-9))

\int \frac{x ^{2} + 9}{x ( x ^{2} - 9 )} d x

- ln ∣ x ∣ + ln ∣ x + 3 ∣ + ln ∣ x - 3 ∣ + C

Solution steps

\int \frac{x ^{2} + 9}{x ( x ^{2} - 9 )} d x

Take the partial fraction of

\frac{x ^{2} + 9}{x ( x ^{2} - 9 )} : - \frac{1}{x} + \frac{1}{x + 3} + \frac{1}{x - 3}

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

\int \frac{1}{x} d x = ln ∣ x ∣

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

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

= - ln ∣ x ∣ + ln ∣ x + 3 ∣ + ln ∣ x - 3 ∣

Add a constant to the solution

= - ln ∣ x ∣ + ln ∣ x + 3 ∣ + ln ∣ x - 3 ∣ + C

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