Partial fractions is a method for rewriting rational expressions containing factors in the denominator as a sum of simpler fractional components. It involves factoring the denominator, setting up partial fractions with unknown constants, and solving equations to determine the constants. This allows integrals of the rational expression to be evaluated using known integration techniques on the individual fractional components. The example worked through shows factoring a denominator, setting up partial fractions with variables A and B, equating coefficients of like terms to set up two equations to solve for A and B, and writing the final partial fraction decomposition. Partial fractions will allow previously difficult integrals to be solved in calculus.

1. PARTIAL FRACTIONS
How to rewrite a rational
expression using partial
fractions

2. PARTIAL FRACTIONS
Is a method for rewriting or “splitting up” a
rational expression that has been simplified
Is going to be a very useful method for us in
Calculus because we will be able to integrate
the partial fractions using techniques we are
familiar with

3. HOW TO FIND PARTIAL FRACTIONS
-Begin by factoring the
denominator:
-Then, set up the partial fraction
decomposition:
-Next, use what we know about
solving rational equations to
eliminate the fractions (multiply
by the LCD):
𝑥−6
𝑥2+4𝑥+3
=
𝑥−6
(𝑥+1)(𝑥+3)
𝑥−6
(𝑥+1)(𝑥+3)
=
𝐴
𝑥+1
+
𝐵
𝑥+3
𝑥 − 6 = 𝐴 𝑥 + 3 + 𝐵(𝑥 + 1)

4. HOW TO FIND PARTIAL FRACTIONS,
CONTINUED
-Since this is an equation, we can
do some creative rearranging:
-Now, since this is an equation,
the left and right sides must be
the same:
-Solve this system of equations
using the method of your choice
(I prefer elimination here)
𝑥 − 6 = 𝐴𝑥 + 3𝐴 + 𝐵𝑥 + 𝐵
𝑥 − 6 = 𝐴 + 𝐵 𝑥 + (3𝐴 + 𝐵)
𝐴 + 𝐵 = 1
3𝐴 + 𝐵 = −6
𝐴 =
9
2
, 𝐵 = −
7
2

5. THE RESULT
𝑥−6
𝑥2+4𝑥+3
=
9
2
𝑥+1
+
−
7
2
𝑥+3
In tomorrow’s announcement, we will look at
how we can now integrate these partial
fractions and have an antiderivative!