Given polynomial , we wish to evaluate integral
It follows that
By the fact (see “Every dog has its day“) that
Let us now consider the case when :
, a polynomial of order .
What emerges from the two cases of is a recursive algorithm for evaluating (1):
Given polynomial ,
Exercise-1 Optimize the above recursive algorithm (hint: examine how it handles the case when )
We will introduce an algorithm for obtaining indefinite integrals such as
or, in general, integral of the form
where is any rational function , with .
Solving (2) for , we have
We also have (see “Finding Indefinite Integrals” )
and (1) is reduced to an integral of rational functions in .
Example-1 Evaluate .
Example-2 Evaluate .
According to CAS (see Fig. 1),
However, the two results are equivalent as a CAS-aided verification (see Fig. 2) confirms their difference is a constant (see Corollary 2 in “Sprint to FTC“).
Exercise-1 According to CAS,
Show that it is equivalent to the result obtained in Example-1
and of course,