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8-5 Partial Fractions. Miss Battaglia AP Calculus. Case 1: Denominator contains only linear factors. Factor the denominator

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8 5 partial fractions

8-5 Partial Fractions

Miss Battaglia

AP Calculus

case 1 denominator contains only linear factors
Case 1: Denominator contains only linear factors
  • Factor the denominator
  • Break up the fraction on the right into a sum of fractions, where each factor of the denominator in Step 1 becomes the denominator of a separate fraction. Then put unknowns in the numerator of each fraction.
  • Multiply both sides of this equation by the denominator of the left side.
  • Take the roots of the linear factors and plug them –one at a time-into x in the equation from Step 3, and solve for unknowns.
  • Plug these results into the A and B in the equation from Step 2.
  • Split up the original integral into the partial fractions from Step 5 and you’re home free!
case 2 the denominator contains irreducible quadratic factors
Case 2: The denominator contains irreducible quadratic factors
  • Factor the denominator
  • Break up the fraction into a sum of “partial fractions”
  • Multiply both sides of this equation by the left-side denominator.
  • Take the roots of the linear factors and plug them-one at a time- into x in the equation from Step 3, and then solve.
  • Plug into Step 3 equation the known values of A and B and any two values for x not used in Step 4 to get a system of two equations in C and D.
  • Solve the system of equations.
  • Split up the original integral and integrate.
distinct linear functions
Distinct Linear Functions

Write the partial fraction decomposition for

example
Example

Write the partial fraction decomposition for

example1
Example

Write the partial fraction decomposition for

classwork homework
Classwork/Homework
  • Read 8.5 Page 561 #7-15