Partial fractions
Videos, Math Dr. Bob:
- Partial Fractions 01 - Distinct Linear Factors
- Partial Fractions 02 - Repeated Linear Factors
- Partial Fractions 03 - Distinct Mixed Factors
- Partial Fractions 04 - Repeated Quadratic Factors
- Partial Fractions 05 - Composition with
01 Theory
A rational function is a ratio of polynomials, for example:
Partial fraction decomposition
The partial fraction decomposition of a rational function is a way of writing it as a sum of simple terms, like this:
Allowed denominators:
- Linear, as any
, or linear power, as any - Quadratic, as any
, or quadratic power, as any
- Condition: quadratics must be irreducible (no roots, or i.e.
) Allowed numerators: constant (over linear power) or linear (over quadratic power)
These are allowed as simple terms in partial fraction decompositions:
These are not allowed:
These are allowed, showing irreducible quadratic and higher powers:
In this example the numerator is linear and the denominator is quadratic and irreducible.
To create a partial fraction decomposition, follow these steps:
- Check denominator degree is higher
- Else do long division
- Factor denominator completely (even using irrational roots)
- Write the generic sum of partial fraction terms with their constants
Repeated factors – special treatment: sum with incrementing powers
- Solve for constants
02 Illustration
Partial fractions with repeated factor
09 - Partial fractions with repeated factor
Find the partial fraction decomposition:
Solution
Check! Numerator is smaller than denominator (degree-wise).
Factor the denominator.
Rational roots theorem:
is a zero. Divide by
: Factor again:
Final factored form:
Write the generic PFD.
Allow all lower powers:
Solve for
, , and . Multiply across by the common denominator:
For
, set , obtain: For
, set , obtain: For
, insert prior results and solve. Plug in
and : Now plug in another convenient
, say :
Plug in
, , for the final answer. Final answer:
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03 Theory
Partial fractions can be integrated using just a few techniques. We consider three kinds of terms:
Linear power bottom
To integrate terms like this:
If
then use log: If
then use power rule:
Quadratic bottom, constant top
We have a formula for simple irreducible quadratics:
This formula should be memorized!
Quadratic bottom, linear top
To integrate terms like this:
Break into separate terms:
Then:
- First term with
in top:
- Second term lacking
in top:
Extra - Completing the square when “no real roots”
To integrate terms with more general quadratics, like this:
we need
, i.e. “no real roots” of the quadratic. If that holds, then we can complete the square and substitute as follows. Look what happens when completing the square:
Notice that
is equivalent to the condition . Create a new label . So this condition means and we can safely define . Then a
-substitution simplifies the equation like this: The quadratic formula
shows that the condition is equivalent to the condition “no real roots.” (In our case . If we had , we could divide out this and change the others.) So we see that “no real roots” is equivalent to the condition that the denominator can be converted to the format
with some constant . At this point, to compute the integral, do trig sub with
and :
04 Illustration
Example - Repeated quadratic, linear tops
10 - Partial fractions - repeated quadratic, linear tops
Compute the integral:
Solution Compute the partial fraction decomposition.
Check that numerator degree is lower than denominator.
Factor denominator completely.
(No real roots.) Write generic PFD:
Notice “linear over quadratic” in first term.
Notice repeated factor: sum with incrementing powers up to 2.
Common denominators and solve:
Therefore:
Integrate by terms.
Integrate the first term using
: Break up the second term:
Integrate the first term of RHS:
Integrate the second term of RHS:
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Extra - “Rationalize a quotient” - convert into PFD
Sometimes an integrand may be converted to a rational function using a substitution.
Consider this integral:
Set
, so and : Now this rational function has a partial fraction decomposition:
It is easy to integrate from there!
Exercise examples:
- To compute
, try the substitution . - To compute
, try the substitution . - To compute
, try the substitution .
Simpson’s Rule
Videos, Math Dr. Bob:
Videos, Organic Chemistry Tutor:
05 Theory - review
The Trapezoid Rule is a technique to approximate the area under a curve as the sum of areas of thin trapezoids whose top corners lie on the curve.
The tops of the trapezoids are lines that approximate the curve. They are determined as lines that agree with the curve at two points.
Trapezoid rule - area formula
Given a function
and a partition of the range labeled by (with and ), the Trapezoid Rule determines the area formula:
Notice the pattern in
Extra - Trapezoid rule - error bound
The error of the Trapezoid Rule approximation is bounded by this formula:
Here
is any number satisfying for .
The Midpoint Rule is a technique to approximate the area under a curve as the sum of areas of thin rectangles whose top midpoints lies on the curve.
The very same formula also represents the areas of trapezoids whose top midpoints lie on the curve and whose top line is tangent to the curve:
The reason they are equal is simple: when pivoting the top line on the ‘attached’ midpoint, the area of the trapezoid does not change.
Midpoint Rule - area formula
Given a function
and a partition of the range labeled by (with and ), the Midpoint Rule determines the area formula: Here each
is the midpoint of the interval . It can be given by the formula .
Extra - Midpoint Rule - error bound
The error of the Midpoint Rule approximation is bounded by this formula:
Here
is any number satisfying for .
Notice that
06 Theory
It turns out that the Midpoint Rule and the Trapezoid Rule tend to differ from the exact integral in opposite directions, and the Midpoint Rule tends to be twice as accurate. Therefore we may improve on both of them by constructing a weighted average of the formulas. This is called Simpson’s Rule.
Simpson’s Rule - defining formula
Simpson’s Rule is given by the weighted sum of the Trapezoid and Midpoint Rules:
Simpson’s Rule - computing formula
Given a function
and a partition of the range labeled by (with and ), Simpson’s Rule determines the area formula:
Note the pattern for Simpson’s Rule: 1, 4, 2, 4, 2, 4, 2,
, 1
Simpson’s Rule - error bound
The error of Simpson’s Rule approximation is bounded by this formula:
Here
is any number satisfying for .
Simpson’s Rule
“Parabola Rule” The formula of Simpson’s Rule can also be explained or defined geometrically: it is the formula giving the sum of areas under small parabolas that meet the curve in three points.
There is a unique parabola passing through any three points with differing
-values: These may be pieced together to form an approximation to the curve:
The area under the parabola through
, , and is given by this formula: This formula may be verified using basic calculus (area under a parabola) and a lot of algebra. (Ambitious students should derive it.)
The area under the parabola through
, , and is given by a similar formula: The Simpson’s Rule formula is the sum of all these formulas! So the
s in Simpson’s come from duplication of endpoint terms as the “rectangular” regions are stacked end-to-end.
07 Illustration
Example - Simpson’s Rule on the Gaussian Distribution
11 - Simpson’s Rule on the Gaussian distribution
The function
is very important for probability and statistics, but it cannot be integrated analytically. Apply Simpson’s Rule to approximate the integral:
with
and . What error bound is guaranteed for this approximation? Solution We need a table of values of
and :
These can be plugged into the Simpson Rule formula to obtain our desired approximation:
To find the error bound we need to find the smallest number we can manage for
. Take four derivatives and simplify:
On the interval
, this function is maximized at . Use that for the optimal : Finally we plug this into the error bound formula:
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