What is the integral of 1/((x-5)(x-4)) ?

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

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integral of 1/((x-5)(x-4))

\int \frac{1}{( x - 5 ) ( x - 4 )} d x

ln ∣ x - 5 ∣ - ln ∣ x - 4 ∣ + C

Solution steps

\int \frac{1}{( x - 5 ) ( x - 4 )} d x

Take the partial fraction of

\frac{1}{( x - 5 ) ( x - 4 )} : \frac{1}{x - 5} - \frac{1}{x - 4}

= \int \frac{1}{x - 5} - \frac{1}{x - 4} 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 - 5} d x - \int \frac{1}{x - 4} d x

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

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

= ln ∣ x - 5 ∣ - ln ∣ x - 4 ∣

Add a constant to the solution

= ln ∣ x - 5 ∣ - ln ∣ x - 4 ∣ + C

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