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