Techniques of Integration: Substitution-Example 2
Techniques of Integration: Substitution-Example 2
Evaluate
Solution. It is clear by looking at the
given integral, that the major problem will be to handle 
and
. Indeed, it is always harder to handle the nth-root
functions when it comes to integration. Therefore a good substitution
will be to take care of both root-functions at the same time. For
example, we may choose 
which will make 
and 
polynomial functions of u. For example, one
may take n = 36 which will give
Clearly we are generating a rational function of u which will take a lot of work to handle. This leads us to reconsider the choice of n and try to make it as small as possible. The right choice will be n=6 which the least common factor of 2 and 3. In this case we have
The method of partial fractions gives
which gives
Hence we get
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