What is the integral of 1/(x^2(x-3)) ?

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

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integral of 1/(x^2(x-3))

\int \frac{1}{x ^{2} ( x - 3 )} d x

- \frac{1}{9} ln ∣ x ∣ + \frac{1}{3 x} + \frac{1}{9} ln ∣ x - 3 ∣ + C

Solution steps

\int \frac{1}{x ^{2} ( x - 3 )} d x

Take the partial fraction of

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

= \int - \frac{1}{9 x} - \frac{1}{3 x ^{2}} + \frac{1}{9 ( 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}{9 x} d x - \int \frac{1}{3 x ^{2}} d x + \int \frac{1}{9 ( x - 3 )} d x

\int \frac{1}{9 x} d x = \frac{1}{9} ln ∣ x ∣

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

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

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

Simplify

= - \frac{1}{9} ln ∣ x ∣ + \frac{1}{3 x} + \frac{1}{9} ln ∣ x - 3 ∣

Add a constant to the solution

= - \frac{1}{9} ln ∣ x ∣ + \frac{1}{3 x} + \frac{1}{9} ln ∣ x - 3 ∣ + C

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