In this class, we continue studying integration techniques. Much as Substitution was related to the Chain Rule, today’s topic – Integration by Parts – is related to the Product Rule.
Basic Learning Objectives
These are the tasks you should be able to perform with reasonable fluency when you arrive at our next class meeting. Important new vocabulary words are indicated in italics.
Given the derivative of a product of functions, find the general antiderivative.
State the rule for Integration by Parts and use it in simple cases.
Advanced Learning Objectives
In addition to mastering the basic objectives, here are the tasks you should be able to perform after class, with practice:
Proficiently use Integration by Parts to calculate indefinite and definite integrals.
Use Integration by Parts in unusual cases, such as \(\int \arctan(x) \; dx\) or \(\int \ln(x) \; dx\).
To prepare for class
Do the first part of the Preview Activity for this section (on WeBWorK if required by your teacher).
Watch the following videos which explain the concept of integration by parts.
Notice that we can write the formula \(\int f(x)g'(x)\;dx = f(x)g(x) - \int g(x)f'(x)dx\) in a more convenient form by replacing \(f(x)\) by \(u\) and \(g(x)\) by \(v\). Using differentials, we also have \(du = f(x)\;dx\) and \(dv = g(x)\;dx\). This finally gives \(\int uv\;dv = uv - \int v\;du.\)
It is important to notice that this new way of writing integration by parts is just a good way to memorize and apply the formula, but in no way are we making any form of substition. In particular, unlike the substitution rule, the bounds of a definite integral where integration by parts is applied won't change.
Do the remaining parts of the Preview Activity for this section (on WeBWorK if required by your teacher).
I am adding more videos here in case you feel the need to see more examples, but mostly these might be videos you may want to look at when reviewing this section for the next exam.
Watch this short video by David Lippman which explains the "LIATE" rule of thumb of choosing \(u\) and \(dv\):