# How do I find the integral #int(x-9)/((x+5)(x-2))dx# ?

##### 1 Answer

This integral can be solved by using the Partial Fractions approach, giving an answer of

#2ln(x+5)-ln(x-2) + C#

**Process:**

The partial fractions approach is useful for integrals which have a denominator that can be factored but not able to be solved by other methods, such as Substitution. This equation already has its denominator factored, but note that if we were instead given the multiplied form:

#int(x-9)/(x^2+3x-10)# ,

we would need to factor the denominator to continue. We can now turn this function into its partial fraction equivalent:

#A/(x+5) + B/(x-2)# =#(x-9)/((x+5)(x-2))#

Multiplying by the common denominator:

#A(x-2) + B(x+5) = x-9#

Now we can choose any value of

Plugging in

#A(-5-2)+B(-5+5) = -5-9#

#A(-7) = -14#

#A = 2#

With

#A(2-2) + B(2+5) = 2 - 9#

#B(7) = -7#

#B = -1#

Using these values in our original partial fractions representation from above, we have:

#A/(x+5) + B/(x-2)# =#2/(x+5) - 1/(x-2)#

Now you can integrate these terms separately using substitution. Both will end up being

#2ln(x+5) - ln(x-2) + C#