As you have seen countless times already, differentiation and integration are intrinsically linked, and for every derivative rule, there is a kindred integral rule.
Formula for Integration by Parts
For example, the chain rule for differentiation corresponds to u-substitution for integration, and the product rule correlates with the rule for integration by parts.
Cool!
Here’s the basic idea. Whenever we have an integral expression that is a product of two mutually exclusive parts, we employ the Integration by Parts Formula to help us evaluate.
What do I mean by mutually exclusive?
It’s when our integral expression is a product where one part is not the derivative of the other.
Now the key to using the integration by parts formula is to correctly choose the right “u” term.
Deciding on a Choice for u
Want to know the secret for choosing the right u-term every time?
It’s the acronym, ILATE!
Get it? “I Late” … I’m late. Haha.
Together, armed with this silly acronym, we will successfully determine the right “u” and “dv” terms, and see how derivative and the integral work together to simplify our expressions.
Thus allowing us to transform the antiderivative of a product of functions into an antiderivative for which a solution can be more easily found, as Wikipedia accurately states.
But, it is important to keep in mind that some integrands will at first glance look like we will need to use integration by parts, but the substitution rule (u-sub) is actually more suitable. So it’s paramount that we slow down and investigate our integral expression, so we choose the right integration technique before jumping in.
Finally, we will see examples of how to use Integration by Parts for Indefinite and Definite Integrals, and learn when we would have to use Integration by Parts more than once, as well as how to use a really nifty technique called the Tabular Method (Tic-Tac-Toe Method) for specific cases.
Integration by Parts Video
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