Some tricky Integration by parts questions

Integration by parts is one of the powerful techniques for solving integrals involving products of functions. We apply this technique whenever the U-substitution fails to solve such products.By playing some tricks with the product rule for derivatives,

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We obtain the integration by parts formula;

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There are some combinations that are very tricky to solve. These include :

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Let’s consider how to evaluate the first two then you try the last one yourself.

Evaluate:

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solution

Let

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and

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This is a u-substitution on its own. Since we used u already, let’s make it a w-substitution

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This is another by-parts on its own. Here is its value:

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Back to main work:

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Let’s verify our solution by evaluating the definite integral:

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Now let’s continue with the second:

Evaluate:

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Solution

Let clip_image002[24]

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And

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clip_image002[28]

clip_image004[18]

clip_image006[10]

clip_image008[8]

clip_image010[6]

But

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clip_image006[12]

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clip_image010[8]

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