# 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

which gives

Hence we get

[Calculus] [Substitution] [More Examples]
[Geometry] [Algebra] [Trigonometry ]
[Differential Equations] [Complex Variables] [Matrix Algebra]

Do you need more help? Please post your question on our S.O.S. Mathematics CyberBoard.

Author: Mohamed Amine Khamsi