Unit 7A

Integrals Part 1

Lesson 2: Basic Integration Properties and Techniques

Integration by Substitution

Whereas the comparison technique relied on inspection, integration by substitution is a more procedural method for finding antiderivatives.

Recall the chain rule for differentiation.


Rewrite both sides in terms of integrals.

     
Let «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»u«/mi»«mo»=«/mo»«mi»g«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/math». The entire integral can be written in terms of the variable «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»u«/mi»«/math» instead of the variable «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»x«/mi»«/math».


Since «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»F«/mi»«mo mathvariant=¨italic¨»`«/mo»«mo»=«/mo»«mi»f«/mi»«/math», the relationship can be rewritten as follows.

Substitution Rule for Indefinite Integrals If «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»u«/mi»«mo»=«/mo»«mi»g«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/math», then «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8747;«/mo»«mi»f«/mi»«mfenced»«mrow»«mi»g«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/mrow»«/mfenced»«mo»§#8729;«/mo»«mi»g«/mi»«mo mathvariant=¨italic¨»`«/mo»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mi»d«/mi»«mi»x«/mi»«mo»=«/mo»«mo»§#8747;«/mo»«mi»f«/mi»«mfenced»«mi»u«/mi»«/mfenced»«mspace width=¨0.125em¨/»«mi»d«/mi»«mi»u«/mi»«/math».

Watch the video Integration by Substitution to see how the substitution method is applied.

As demonstrated in the video, there are 5 steps to follow when applying the method of integration by substitution. These are listed below.

Steps for Integrating by Substitution