Integration by Parts: Example 2

Evaluate

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First let us point out that we have a definite integral. Therefore the final answer will be a number not a function of x! Since the derivative or the integral of tex2html_wrap_inline24 lead to the same function, it will not matter whether we do one operation or the other. Therefore, we concentrate on the other function tex2html_wrap_inline26 . Clearly, if we integrate we will increase the power. This suggests that we should differentiate tex2html_wrap_inline26 and integrate tex2html_wrap_inline24 . Hence

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After integration and differentiation, we get

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The integration by parts formula gives

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It is clear that the new integral tex2html_wrap_inline38 is not easily obtainable. Due to its similarity with the initial integral, we will use integration by parts for a second time. The same discussion as before leads to

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which implies

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The integration by parts formula gives

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Since tex2html_wrap_inline46 , we get

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which finally implies

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Easy calculations give

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From this example, try to remember that most of the time the integration by parts will not be enough to give you the answer after one shot. You may need to do some extra work: another integration by parts or use other techniques,....

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Author: Mohamed Amine Khamsi

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