What is the integral of 1/((x-5)(x-3)) ?

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

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integral of 1/((x-5)(x-3))

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

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

Solution steps

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

Take the partial fraction of

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

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

\int \frac{1}{2 ( x - 5 )} d x = \frac{1}{2} ln ∣ x - 5 ∣

\int \frac{1}{2 ( x - 3 )} d x = \frac{1}{2} ln ∣ x - 3 ∣

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

Add a constant to the solution

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

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