What is meant by partial fractions with quadratic denominators?

• For?linear?denominators the denominator of the original fraction can be factorised such that the denominator becomes a product of linear terms of the form
• With?squared?linear?denominators, the same applies, except that some (usually just one) of the factors on the denominator may be squared, i.e.
• In both the above cases it can be shown that the?numerators?of each of the partial fractions will be a?constant
• For this course,?quadratic?denominators?refer to fractions that contain a?quadratic?factor?(that?cannot?be?factorised) on the denominator
  • the denominator of the quadratic partial fraction will be of the form ; very often ?leaving it as
  • the numerator of the quadratic partial fraction could be of linear form,

How do I find partial fractions involving quadratic denominators?

• ?STEP 1??????????Factorise?the denominator as far as possible (if not already done so)

• • Sometimes the numerator can be factorised too
• STEP 2??????????Split?the fraction into a?sum?with
  • the?linear?denominator?having an (unknown)?constant?numerator
  • the?quadratic?denominator?having an (unknown)?linear?numerator
• STEP 3????????? Multiply through by the denominator to eliminate fractions

• STEP 4????????? Substitute values into the identity and solve for the unknown constants
  • Use the root of the?linear?factor as a value of ?to find one of the unknowns
  • Use any two values for ?to form two equations to solve simultaneously
    • is a good choice if this has not already been used with the linear factor
• STEP 5????????? Write the?original?as partial fraction

How do I integrate the fraction with the quadratic denominator?

• ?The quadratic denominator will be of the form
  • If it is not then you can get it to look like this by?completing the square
• Split into to fraction
• Integrate ?using logarithms to get
• Integrate ?using the?formula booklet?or using a?trigonometric or hyperbolic?substitution
  • If?a?and?c?have the same sign then use
  • If?a?and?c?have different signs then use
    • Or in this case you can factorise using surds and then use partial fractions