The trigonometric functions () and their hyperbolic equivalents () can be used in integrals through the use of their identities. The process can be thought of as a reverse U-Substitution.
Definition
Just as u-substitution uses the rule: We can ‘reverse’ the rule and perform another change of variables to simplify the integral:
(where is the inverse function of )
The following table helps quickly identify what type of trigonometric/hyperbolic substitution is needed, given a certain integral form.
Integrand | Substitution |
---|---|
Examples
1: Cosh Substitution
Solve the following integral using a trigonometric or hyperbolic substitution:
The integral is of the form . so we’ll be using a cosh substitution: Let . Then The integral can now be expressed in terms of : /