Integration by Parts… How? (NancyPi)

MIT grad shows how to integrate by parts and the LIATE trick. To skip ahead: 1) For how to use integration by parts and a good RULE OF THUMB for CHOOSING U and DV, skip to time 2:46. 2) For the TRICK FOR CHOOSING U and DV (the LIATE memory trick) skip to 10:12. Nancy formerly of MathBFF explains the steps.

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WHEN to use INTEGRATION BY PARTS: If you have an integral to evaluate, and you don’t already know how to integrate it, as is, then see if you can simplify it somehow with algebra. If not, try doing a substitution, like U-Substitution. If U-Substitution does not help, then you may need to use the INTEGRATION BY PARTS formula.

RULE OF THUMB: The first step to use Integration by Parts is to pick your “u” and “dv”. As a general rule of thumb, whichever factor in your integrand gets simpler when you take the derivative of it, make that your u. Then make the other factor your dv (and include the dx in this dv). Note: This works if the part you chose for dv does not get any bigger or more complicated when you integrate it. After you’ve picked u and dv, then find du by differentiating u, and find v by integrating dv. Finally, plug everything into the integration by parts formula and simplify.

TRICK: You can instead use an acronym memory trick to choose the right “u”. The acronym LIATE stands for Logs, Inverse trig functions, Algebraic functions, Trig functions, and Exponentials (in that order). If you follow those letters, LIATE, in sequence, whichever type of function you find first that you have, make that your u. Then, make the next thing you find your dv. Once you’ve picked u and dv this way, the steps are the same as above. Find du by differentiating u, and find v by integrating dv. Then use the IBP (integration by parts) formula and simplify for the answer.

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