University:
Edmonds College
Course:
MATH 077 | Introductory Algebra
Academic year:

2023
Views:

331

Pages:

5
Author:

Ramy M.

Partial Fractions ( ) A rational function ( ) ( ) can be expressed as a sum of simpler fractions, called Partial Fractions. 1. Improper rational function degree of ( ) degree of ( ) In the case, we must first perform long division to rewrite the quotient ( ) ( ) in the form ( ) ( ) . ( ) Example Transform the following fraction into partial fraction ( ) Solution by using long division )̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ ( ) 2. Proper rational function degree of ( ) degree of ( ) There are different cases for proper rational function. ( ) is a product of distinct linear factors Case I ( ) ( ( ) ) )( ( ) Then, to this factor, assign the sum of the ( ) ( ) ( partial fractions: ) ( ) Example Transform the following fraction into partial fraction , and then find ∫ ( ) ( ) Solution Method 1 Solve by substitution ( ( )( ) ( ) ( ( ) ( ) ( ) ( )( ) ) Put Put ( ) ( ) ( ) Method 2 Solve by equating corresponding coefficients Coefficients of (1) Coefficients of (2) solving two equations, we get ∫ (( ) ( ) ) | | | | ) Case II ( ) is a product of linear factors, some of which are repeated ( ) ( ( ) ) )( ( ) Then, to this factor, assign the sum of the ( ) ( ) ( partial fractions: ) ( ) Example Transform the following fraction into partial fraction ( ) ( ,and then find ∫ ( ) ) Solution ( ( ) ( ) ( ) ( ) ( ) ( ( ) ( ) ) ) Put Put (1) Put (2) Substitute from equation (2) in equation (1) and ( ) ∫ ( ) ∫ (( | ) ( ( | ) ) ( ( ( ) ) ) ) ( ( ) ) Case III ( ) is irreducible quadratic factors and non-repeated ( ) ( ) ( ) Then, to this factor, the partial fraction is: ( ) ( ) Example Transform the following fraction into partial fraction ( ) ( )( ) , and then find ∫ ( ) Solution ( )( ) ( ( )( )( ) ( ∫ (( ) ) ) ( ) ( ) Put Put Put ( ∫ (( ∫ (( ∫( ) ) ) ) ) ∫( ) = ∫ ( ) |( ) ) ( ) Split up the integral ∫( ) )| ….(i) and ∫ ∫( ( Substitute ) and ∫ ( ( ) ) ….(ii) ( ∫ ) | ) …..(iii) | add equations (i), (ii) and (iii) ∫ ( )( ) |( )| ( ) | | Remember Try to solve Transform the following fraction into partial fraction 1- 2- 3- 4- 5- ( 6- ) 79- 8( ) 10- ( ) ( )