6.5 Partial Fraction Decomposition

Question 1

when do you use fration decomposition

Answer 1

when your integral is a polynomial divided by a polynomial

Question 2

what are the two scenarios of fraction decomposition

Answer 2

being divided by x-a to some power

or being divided by x^2 +bx + c to some power

Question 3

how to decomp when being divided by x-a to some power

Answer 3

separate into the powers of x-a until the highest one

ex: x-a to the power of 2

one fraction
A / x-a

other fraction
B / x-a to the power of 2

Question 4

how to do decomp when being divided by x^2+bx+c to some power

Answer 4

separate into the powers of x^2+bx+c until the highest one with the numerators being Ax+ b and so on

ex: x^2+bx+c to the power of 2

one fraction
Ax + B / x^2+bx+ c

other fraction
Cx + D / x^2+bx+c to the power of 2

Question 5

steps for solving a decomp question

Answer 5

1. separate the given integrand
2. multiply by the original denominator to both sides
3. simplify my multiplying out
4. move terms next to like terms and factor out the like term (x)
5. set the things being multiplied by the like term of x to its corresponding number on the other side of the equation
6. solve for numerator variables with a systems of equations
7. plug variables into the first made separated equation
8. rewrite integral and solve now that it is easier