What kinds of functions can be integrated using partial fraction decomposition?

Problem 1E

Solution:-

Step1

we will assume that we have a rational function in which degree of p (x) < degree of q (x). If this is not the case, we can always perform long division.

For example, if we were given the fraction .

We would perform long division to obtain

=x+

We then would apply partial fraction decomposition to .

Step2

(1) q(x) is a product of distinct linear factors.

Let us assume that q (x) is a product of n distinct linear factors that is

q (x) =() ):::()

Then,

++-------------+

Finding the decomposition amounts to finding the coefficient A1, ..., An. This can be done two different ways.

Step3

(2)q(x) is a product of linear factors, some being repeated

The factors which are not repeated will be decomposed as above.

Suppose that q (x) also contains that is ax + b is repeated m times.

The decomposition for this factor will be

+