What is the integral of 1/((x-4)(x+2)) ?

The integral of 1/((x-4)(x+2)) is 1/6 ln|x-4|-1/6 ln|x+2|+C

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integral of 1/((x-4)(x+2))

\int \frac{1}{( x - 4 ) ( x + 2 )} d x

\frac{1}{6} ln ∣ x - 4 ∣ - \frac{1}{6} ln ∣ x + 2 ∣ + C

Solution steps

\int \frac{1}{( x - 4 ) ( x + 2 )} d x

Take the partial fraction of

\frac{1}{( x - 4 ) ( x + 2 )} : \frac{1}{6 ( x - 4 )} - \frac{1}{6 ( x + 2 )}

= \int \frac{1}{6 ( x - 4 )} - \frac{1}{6 ( 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}{6 ( x - 4 )} d x - \int \frac{1}{6 ( x + 2 )} d x

\int \frac{1}{6 ( x - 4 )} d x = \frac{1}{6} ln ∣ x - 4 ∣

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

= \frac{1}{6} ln ∣ x - 4 ∣ - \frac{1}{6} ln ∣ x + 2 ∣

Add a constant to the solution

= \frac{1}{6} ln ∣ x - 4 ∣ - \frac{1}{6} ln ∣ x + 2 ∣ + C

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What is the integral of 1/((x-4)(x+2)) ?
The integral of 1/((x-4)(x+2)) is 1/6 ln|x-4|-1/6 ln|x+2|+C