Integration by Parts: Example 3

Find

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The two functions involved in this example do not exhibit any special behavior when it comes to differentiating or integrating. Therefore, we choose one function to be differentiated and the other one to be integrated. We have

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which implies

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The integration by parts formula gives

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The new integral tex2html_wrap_inline213 is similar in nature to the initial one. One of the common mistake is to do another integration by parts in which we integrate tex2html_wrap_inline215 and differentiate tex2html_wrap_inline217 . This will simply take you back to your original integral with nothing done. In fact, what you would have done is simply the reverse path of the integration by parts (Do the calculations to convince yourself). Therefore we continue doing another integration by parts as

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which implies

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Hence

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Combining both formulas we get

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Easy calculations give

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After two integration by parts, we get an integral identical to the initial one. You may wonder why and simply because the derivative and integration of tex2html_wrap_inline229 are the same while you need two derivatives of the cosine function to generate the same function. Finally easy algebraic manipulation gives

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Try to find out how did we get the constant C?

In fact we have two general formulas for these kind of integrals

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and

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Author: Mohamed Amine Khamsi

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