Calculus II Workbook For Dummies
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Sometimes you have to use the integration-by-parts method more than once because the first run through the method takes you only part way to the answer.

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  1. Go down the LIATE list and pick your u.

    Using the mnemonic device LIATE:

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    To pick your u, go down this list in order; the first type of function on this list that appears in the integrand is the u. Then everything else is automatically the dv.

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  2. Organize the problem using the first box shown in the figure below.

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    In the second box above, your u and dv from Step 1 have been put in the u and dv cells. The third box shows the completed breakdown of the problem after differentiating the u and integrating the dv.

  3. Use the integration-by-parts formula.

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    You end up with another integral,

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    that can’t be done by any of the simple methods — reverse rules, guess and check, and substitution. But note that the power of x has been reduced by one, so you’ve made some progress.

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  4. Integrate by parts again.

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    (If you feel like it, you can change the negative C into a positive C because C is just any old constant.)

  5. Take the result from Step 4 and substitute it for the

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  6. in the answer from Step 3 to produce the whole enchilada.

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(In this last line, you can replace the 2C with C because twice any old constant is the same thing as any old constant. This looks weird mathematically, but, don’t worry, it’s okay.)

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