Techniques of Integration: Substitution-Example 2

Evaluate

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Solution. It is clear by looking at the given integral, that the major problem will be to handle tex2html_wrap_inline70 and tex2html_wrap_inline72 . 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 tex2html_wrap_inline74 which will make tex2html_wrap_inline76 and tex2html_wrap_inline78 polynomial functions of u. For example, one may take n = 36 which will give

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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

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The method of partial fractions gives

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which gives

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Hence we get

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Author: Mohamed Amine Khamsi

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