Examples

Partial fractions with repeated factor

Find the partial fraction decomposition:

Solution

(1) Check that denominator degree is lower.

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(2) Factor denominator:

Rational Roots Theorem: check for roots at and and .

Discover that is a root. Therefore divide by :

Factor again:

Final factored form:

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(3) Write the generic PFD:

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(4) 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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(4) Plug in , , for the final answer:

Partial fractions - repeated quadratic, linear tops

Compute the integral:

Solution

(1) Partial fraction decomposition:

• Numerator degree is lower than denominator.
• Factor denominator completely. (No real roots.)

Write generic PFD:

• Notice: repeated factor: use incrementing powers up to 2.
• Notice: linear over quadratic.

Common denominators and solve:

Therefore:

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(2) Integrate:

Integrate the first term using substitution :

Break up the second term:

Integrate the first term of RHS:

Integrate the second term of RHS using :