Sec. 5.4 - Integration by Parts

Overview

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 your 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 only part (a) of the Preview Activity for this section (on WeBWorK if required by your teacher).

• Watch the following video (by GVSUmath) which explains the concept of integration by parts:

  Two ways to write 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 then have \(du = f(x)\;dx\) and \(dv = g'(x)\;dx\). This finally gives

  $$\int u\;dv = uv - \int v\;du.$$

  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.

• Watch the following videos (by GVSUmath) which show examples of applying the method of integration by parts:

• Watch the following video (by GVSUmath) which shows how to use integration by parts for a definite integral directly:

• Do the remaining parts of the Preview Activity for this section (on WeBWorK if required by your teacher).

After class

• Watch this short video (by David Lippman) which explains the “LIATE” rule-of-thumb of choosing \(u\) and \(dv\):

• Watch the following videos (by GVSUmath and Nakia Rimmer, respectively) with more example calculations as needed, in particular when reviewing this section for the next exam:

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