W03 - Examples

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:

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Write the generic PFD.

Allow all lower powers:

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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 :

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Plug in , , for the final answer.

Final answer:

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:

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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:

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: