6.3 Partial Fractions

Question 1

Partial Fractions are used to integrate…

Answer 1

rational functions (ratios of polynomials)

Question 2

Prerequisites to partial fraction decomposition

Answer 2

1. Degree of numerator

Question 3

Each linear factor (ax+b) gives a term of the form…

Answer 3

A/(ax+b), A is a constant

Question 4

Each repeated linear factor of the form (ax+b)ⁿ gives a series of terms of the form

Answer 4

A₁ / (ax+b) + A₂ / (ax+b)² … An₋₁ / (ax+b)ⁿ⁻¹ + An / (ax+b)ⁿ

Question 5

Each irreducible quadratic factor of the form (ax² + bx + c) gives a term of the form

Answer 5

(Bx+C)/(ax²+bx+c) B and C are constant

Question 6

Each repeated quadratic factor of the form (ax² + bx + c)ⁿ gives a series of terms of the form

Answer 6

(B₁x+C₁)/(ax²+bx+c) + (B₂x+C₂)/(ax²+bx+c)² … + (Bn₋₁x+Cn₋₁)/(ax²+bx+c)ⁿ⁻¹ + (Bnx+Cn)₁/(ax²+bx+c)ⁿ

Question 7

Steps of partial fraction decomposition

Answer 7

1. Split into terms based on terms in denominator
2. Determine the coefficients in the numerators by cross-multiplying by the LCD on the right and setting the numerators on the right equal to the one on the left
3. Integrate each term

Question 8

The two methods of determining partial fraction coefficents

Answer 8

1. Sub in roots as values of x

2. expand the right side, collect like terms, and match the powers of x