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Partial Fractions

The challenge in this section is to evaluate

integrals of the form

such as

Partial Fractions

- a technique that decompose a fraction or a

rational function into a sum of several fractions

(or rational functions) with simpler denominators.

Examples

and

etc.

Review A rational function is of the form

where p(x), q(x) are polynomials in x and q(x) is

nonzero.

We say that the rational function

is reduced

if the degree of p(x) is less than that of q(x).

Assuming that

is reduced, we will consider

several cases

Case 1 q(x) is a product of distinct linear

factors, eg. q(x) (x -

a)(x - b)(x - c)

where a, b, c are all different.

In this case,

for some constants A, B, and C that can be

determined by substitution.

Case 2 q(x) is a product of repeated linear

factors, eg. q(x) (x -

a)(x - b)(x - b)

where a, b are different.

In this case,

for some constants A, B, and C that can be

determined by substitution or comparing

coefficients.

Case 3 q(x) is a product of linear and

irreducible distinct quadratic factors,

eg. q(x) (x - a)(x - b)(cx2 dx e)

where a,

b are different.

Irreducible means that (cx2 dx e) does not

have real roots.

In this case,

for some constants A, B, C, and D that can be

determined by substitution or comparing

coefficients.

Case 3 q(x) is a product of linear and

irreducible distinct quadratic factors,

To integrate rational expressions of the form

we have to complete the square in the denominator

and then integrate by substitution. This

technique is best explained by examples.

Example

We first need to perform some algebra before

integrating

(No Transcript)

The first expression can be integrated by

substituting u (

x2 2x 5) while the second one can be

integrated by substituting

( x 1) 2 tan

(No Transcript)

The final answer is therefore

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