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

The integral of 1/((x-2)(x-3)(x+2)) is -1/4 ln|x-2|+1/5 ln|x-3|+1/20 ln|x+2|+C

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

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

- \frac{1}{4} ln ∣ x - 2 ∣ + \frac{1}{5} ln ∣ x - 3 ∣ + \frac{1}{20} ln ∣ x + 2 ∣ + C

Solution steps

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

Take the partial fraction of

\frac{1}{( x - 2 ) ( x - 3 ) ( x + 2 )} : - \frac{1}{4 ( x - 2 )} + \frac{1}{5 ( x - 3 )} + \frac{1}{20 ( x + 2 )}

= \int - \frac{1}{4 ( x - 2 )} + \frac{1}{5 ( x - 3 )} + \frac{1}{20 ( x + 2 )} 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}{4 ( x - 2 )} d x + \int \frac{1}{5 ( x - 3 )} d x + \int \frac{1}{20 ( x + 2 )} d x

\int \frac{1}{4 ( x - 2 )} d x = \frac{1}{4} ln ∣ x - 2 ∣

\int \frac{1}{5 ( x - 3 )} d x = \frac{1}{5} ln ∣ x - 3 ∣

\int \frac{1}{20 ( x + 2 )} d x = \frac{1}{20} ln ∣ x + 2 ∣

= - \frac{1}{4} ln ∣ x - 2 ∣ + \frac{1}{5} ln ∣ x - 3 ∣ + \frac{1}{20} ln ∣ x + 2 ∣

Add a constant to the solution

= - \frac{1}{4} ln ∣ x - 2 ∣ + \frac{1}{5} ln ∣ x - 3 ∣ + \frac{1}{20} ln ∣ x + 2 ∣ + C

What is the integral of 1/((x-2)(x-3)(x+2)) ?
The integral of 1/((x-2)(x-3)(x+2)) is -1/4 ln|x-2|+1/5 ln|x-3|+1/20 ln|x+2|+C