We will introduce an algorithm for obtaining indefinite integrals such as

or, in general, integral of the form

where is any rational function , with .

Let

Solving (2) for , we have

which provides

and,

yields

Similarly,

gives

We also have (see “Finding Indefinite Integrals” )

.

Hence

,

and (1) is reduced to an integral of rational functions in .

Example-1 Evaluate .

*Solution:*

.

Example-2 Evaluate .

*Solution:*

.

According to CAS (see Fig. 1),

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“).

Fig. 2

Exercise-1 According to CAS,

Show that it is equivalent to the result obtained in Example-1

Exercise-2 Try

and of course,