Let f(x)= 1/(x^2 -4)

We need to determine F(x) such that:

F(x) = intg f(x) dx

==>F(x) = intg [ 1/(x^2 - 4)] dx

Let us simplify:

1/(x^2 -4) = 1/(x-2)(x+2)

==>            = A/(x-2) + B/(x+2)

==> A(x+2) + B(x-2) = 1

Open brackets:

==> Ax  2A + Bx - 2B = 1

==> (A+B) x + 2(A-b) = 1

==> A+b = 0 .....(1)

==> A-b = 1/2......(2)

Now add (1) and (2):

==> 2A = 1/2

==> A = 1/4

==> B = -1/4

==> 1/(x^2 - 4) = 1/4(x-2)  - 1/4(x+2)

==> F(x) = intg [1/4(x-2) - 1/4(x+2) ] dx

                = (1/4)*( intg 1/(x-2) dx - intg 1/(x+2) dx

                = (1/4)* [ ln (x-2) - ln (x+2) ] + C

                = (1/4) ln (x-2)/(x+2)  + C

==> F(x) = (1/4)*ln (x-2)/(x+2)  + C