Change of variable substitution
Definition
A change of variable substitution, called a substitution for brevity, refers to a situation where one variable (or a collection of variables) is replaced by another variable (or collection of variables) where the new variable is typically expressed as a function of the old variable. In many cases of interest, the substitution is (at least locally) reversible, i.e., the old variable(s) can be expressed back in terms of the new variable(s).
Substitution is used in many parts of calculus and its main goal is to reduce one problem to another, possibly simpler, problem.
Substitution methods are usually related to the act of breaking down a function as a composite of two functions, with the inner function is the function being substituted.
Topics that use substitution
| Topic name | Concept of substitution | What is the substitution performed on? |
|---|---|---|
| indefinite integration -- finding the antiderivative of a function also, definite integration |
integration by substitution (also called integration by -substitution) | the variable of integration; in case of definite integration, the domain of integration is also transformed. |
| computing sums of power series | substitution method for power series summation | the variable whose powers are being considered |
| differential equations | substitution method for solving differential equations | we may perform a substitution on the dependent variable or the independent variable, or both |
