Expert: Paul Klarreich - 8/14/2009

Question
Please evaluate the integral of x - 1/x(x + 5) dx using the partial fractions method.. thankyou

Answer
Questioner: heather
Country: Papua New Guinea
Category: Calculus
Private: Yes  <<  DON'T DO THIS.
Subject: Applications of Integration
Question: Please evaluate the integral of x - 1/x(x + 5) dx using the partial fractions method.. thankyou
........................
You wrote:

        1
x -  ---------
    x(x + 5)

But did you mean:

x - 1
----------
x(x + 5)

If so, here is how to handle it:

Write:

x - 1       A       B
-------- =  --- + -------
x(x + 5)     x     x + 5

Multiply through:

x - 1 = A(x + 5) + Bx

Now there are two ways to handle it.  Both involve solving equation for A and B.

1. Choose two nice values for x.  (Two, because you have two things, A, B, to find.)  Write the equations and solve.

2. Multiply out.  Equate:
  Coefficients of x on both sides.
  Constant terms on both sides.
Write the equations and solve.


I'll do 2:

x - 1 = Ax + 5A + Bx

x:   1 = A + B
c:  -1 = 5A.

You will get  A = -1/5,  B = 6/5.

Now you get:

x - 1      -1/5       6/5
-------- =  ------ + -------
x(x + 5)     x        x + 5

and you can do the rest.