Integration, Integration by Using Partial Fractions, Factor in the Denominator, Corresponding Partial Fraction
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Integration by Using Partial Fractions
By now we are equipped with the various techniques of integration.
But there still may be a case like , where the substitution or the integration by parts may not be of much help. In this case, we take the help of another technique called technique of integration using partial functions.
Any proper rational fraction can be expressed as the sum of rational functions, each having a single factor of . Each such fraction is known as partial fraction and the process of obtaining them is called decomposition or resolving of the given fraction into partial fractions.
For example,
Here are called partial fractions of .
If is a proper fraction and can be resolved into real factors then,
(a) Corresponding to each non repeated linear factor , there is a partial fraction of the form
(b) For we take the sum of two partial fractions as
For we take the sum of three partial fractions as and so on.
(c) For non-fractorisable quadratic polynomial there is a partial fraction
Therefore, if is a proper fraction and can be resolved into real factors, can be written in the following form:
Factor in the denominator | Corresponding partial fraction |
Where are arbitary constants?
The rational functions which we shall consider for integration will be those whose denominators can be fracted into linear and quadratic factors.
Example:
Evaluate:
Solution:
Let
Multiplying both sides by , we have
Putting , weget or
Putting , we get or
Substituting these values in, we have