# Calculus 102 - Lecture 12

today we will move on to introduce some new integration
techniques.

Very often, we will need to integrate a very special type of

function which is called rational functions . Here is the definition.

**Definition 1** A rational function is the function of the form

. where P(x) and Q(x) are both **polynomials**.

The way we will apply is called method of partial fraction.

**
Definition 2 (Method of Partial Fraction) **The method of partial

fraction is the algebraic technique that decomposes R(x) into a

sum of terms :

where p(x) is a polynomial and F_{i}(x) is a fraction that
can be integrated

without difficulties.

We know from basic algebra that the only inreducible polynomials

with coefficient in R are quadratic and linear polynomials. Believe

it or now, Using this principle, we can show every rational function

can be written as equation (1), with F_{i}(x) being either

. or

. Fractions of these forms are called** partial fractions**.
Equation(1)

is called **partial fraction decomposition.**

We need to learn to deal with the first type of partial fractions

now. We need a Definition first.

**Definition 3** A fraction is called proper if
and only if the degree

of P(x) is strictly less than that of Q(x).

The harder part is to deal with proper fractions, but now we need

to deal with fraction that is not proper. The technique is called **long
division.**

Please see the following examples.

**Example 1**please find

**Example 2**please find

So I need you to remember now, the very first step to integrate a

rational function is to apply long division to reduce the

**improper**

fraction to proper fraction.

fraction to proper fraction.

.

Then we need to factorize the denominator. As we just said, the only

inreducible polynomial is quadratic and linear polynomial, so you

should express the bottom as the product of linear and quadratic

IRREDUCIBLE polynomials.

The first rule you should apply is

**Rule 1**The part of the partial fraction decompsition that corresponds

to the linear factor ax + b with multiplicity n is

of course the capital A_{i} is constants.

**Example 3 **please find

**Example 4 **please find .

Here you will have to FACTORIZE the denominator yourself .

Let us see another more difficult problem.

**Example 5** please find

The idea is

• you can list the factors according to rule 1,

• and then you need to find the common denominator of these

**partial fractions** which is of course the original denominator(

believe or not),

• then write the partial fractions using this common denominator,

• arrange the terms and match the coefficients with that of the

original numerator, you will get a list of simultaneous equations .

• solve them to get the coefficient A_{i}.

• integrate it.

**Example 6** please find

**Example 7** please find

**Example 8 **please find

**Example 9** please find

**Rule 2** The part of the partial fraction decompsition that
corresponds

to the quadratic factor ax^{2} + bx + c with multiplicity n is

of course the capital Bi and C_{i} is constants.

Here sometimes you will see very complicated examples if you were

given some very large multiplicities for the quadratic factor. I was

tortured that way when I first learnt this. But you can be assured

it will not happen to you.

What you need to pay attention here is that the numerator is a linear

polynomial instead of a constant.

Do not confuse yourself with the linear factor case.

Here is several examples for you to do.

**Example 10**

**Example 11**

**Example 12**

As the book suggests, you might consider using the
following formula ,

the reason I put them here is to help you do problem faster.

But I will assume you can find it yourself when you see them in the

problem.

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