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Solved: Setting up partial fraction decompositions Give

ISBN: 9780321570567 2

Solution for problem 27E Chapter 7.4

Calculus: Early Transcendentals | 1st Edition

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Problem 27E

Setting up partial fraction decompositions

Give the appropriate form of the partial fraction decomposition for the following functions.

Step-by-Step Solution:
Step 1 of 3

Problem 27E

Answer;

Step-1

Proper fraction definition ; In a rational fraction  , if the degree of f(x) < the degree of g(x) , then the rational fraction is called a proper fraction.

The sum of two proper fractions is a proper fraction.

Example;

Partial fractions Depending upon the nature of factors of Denominator ;

When the denominator has non-repeated linear factors;

A non - repeated linear factor (x-a) of denominator corresponds a partial fraction of the form  , where A is a constant to be determined’

If g(x) = (x-a)(x-b)(x-c)............(x-n), then we assume that

=  ++ +...............+

Where A, B, C,............N are constants which can be determined by equating the numerator of L.H.Sto the numerator of R.H.S , and substituting x = a,b ,c ….n.

Step-2

2) When the denominator has repeated  linear factors

A repeated linear factor  of denominator corresponds partial fractions of the form ;

=  + ++......................+

Where A, B, C,............N are constants which can be determined by equating the numerator of L.H.Sto the numerator of R.H.S , and substituting x = a, we get N.

Example;

Step-3

3) When the denominator has non repeated quadratic factors;

To every quadratic factor (which cannot be factored into linear factors) of the form a+bx+c in the denominator , there will be a partial fraction of the form , where A and B are constants to be determined.

Example; = ++

Step-4

The given integral is ; dx………………(1)

Here the given integrand is of the form   , and f(x) < g(x).

Therefore , the given fraction is a proper fraction , and the denominator has  repeated linear factors.

Thus , from the above step the given fraction can be written as;

= + +

= …………..(2)

...

Step 2 of 3

Step 3 of 3

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