Additional Resources
OpenStax Online Textbook
How to Read a Math Textbook
Additional Instructional Resources
The Substitution Rule
Objectives
- Use substitution to evaluate indefinite integrals
- Use substitution to evaluate definite integrals
Summary
To find algebraic formulas for antiderivatives of more complicated algebraic functions, we need to think carefully about how we can reverse known differentiation rules. To that end, it is essential that we understand and recall known derivatives of basic functions, as well as the standard derivative rules.
The indefinite integral provides notation for antiderivatives. When we write "", we mean "the general antiderivative of ". In particular, if we have functions and such that , the following two statements say the exact thing:
That is, is the derivative of , and is an antiderivative of .
The technique of u-substitution helps us evaluate indefinite integrals of the form through substitutions and so that
A key part of choosing the expression in to be represented by is the identification of a function-derivative pair. To do so, we often look for an "inner" function that is part of a composite function, while investigating whether (or a constant multiple of ) is present as a multiplying factor of an integrand.