Lec 6 - Integration by Rational Function by Partial Fraction

Sec 7.4

 

 Techniques of Integration

 

1. Substitution
2. Integration by parts
3. Trig Integrals
4. Trig substitutions
5. Partial functions

 

Recall,




We will learn how to do this backwards so that the integrand can be rewritten as a sum of terms each of which we can be rewritten as a sum of terms each of which we can integrate.

 

Definition





We only apply partial fractions on proper rational functions.

If not proper then we long division to reduce it



 

Example

 

Evaluate



 

We can use long division.

















 



 





 

 

Step 1: Use long division to get a proper fraction

 

Step2: Decompose our proper rational function

 



 





 

In this case we always assume,







 

The method I present is different from the textbook.

 





 





 

On the LHS, we have



 

Therefore,



 

Eg

 

Evaluate



 

Step 1: It is proper, no long division.

Step 2: We use partial fractions

 



 





 

Step 3: Evaluate

 









 

What if weinstead have