What is the integral of 5/((x-9)(x-4)) ?

The integral of 5/((x-9)(x-4)) is ln|x-9|-ln|x-4|+C

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integral of 5/((x-9)(x-4))

\int \frac{5}{( x - 9 ) ( x - 4 )} d x

ln ∣ x - 9 ∣ - ln ∣ x - 4 ∣ + C

Solution steps

\int \frac{5}{( x - 9 ) ( x - 4 )} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 5 \cdot \int \frac{1}{( x - 9 ) ( x - 4 )} d x

Take the partial fraction of

\frac{1}{( x - 9 ) ( x - 4 )} : \frac{1}{5 ( x - 9 )} - \frac{1}{5 ( x - 4 )}

= 5 \cdot \int \frac{1}{5 ( x - 9 )} - \frac{1}{5 ( 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

= 5 (\int \frac{1}{5 ( x - 9 )} d x - \int \frac{1}{5 ( x - 4 )} d x)

\int \frac{1}{5 ( x - 9 )} d x = \frac{1}{5} ln ∣ x - 9 ∣

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

= 5 (\frac{1}{5} ln ∣ x - 9 ∣ - \frac{1}{5} ln ∣ x - 4 ∣)

Simplify

5 (\frac{1}{5} ln ∣ x - 9 ∣ - \frac{1}{5} ln ∣ x - 4 ∣) : ln ∣ x - 9 ∣ - ln ∣ x - 4 ∣

= ln ∣ x - 9 ∣ - ln ∣ x - 4 ∣

Add a constant to the solution

= ln ∣ x - 9 ∣ - ln ∣ x - 4 ∣ + C

What is the integral of 5/((x-9)(x-4)) ?
The integral of 5/((x-9)(x-4)) is ln|x-9|-ln|x-4|+C