Calculus- Techniques of Integration: Substitution-Example1

Find

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Solution. It is clear that once we develop the tex2html_wrap_inline33 through the binomial formula, we will get a polynomial function easy to integrate. But it is clear that this will take a lot of time with big possibility of doing mistakes !!
Let us consider the substitution tex2html_wrap_inline35 (the reason behind is the presence of x in the integral since the derivative of tex2html_wrap_inline39 is 2x). Indeed, we have du = 2x dx and therefore

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We may check that the new integral is easier to handle since tex2html_wrap_inline47 . Hence

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which does not complete the answer since the indefinite integral tex2html_wrap_inline51 is a function of x not of u. Therefore, we have to go back and replace u by u(x):

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Author: Mohamed Amine Khamsi

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