8.
9.
TrigonometricSubstitutions
LearningObjectives
A studentwill be able to:
- Computeby handthe integralsof a wide varietyof functionsby usingtechniqueof TrigonometricSubsti-
tution. - Combinethis techniquewith otherintegrationtechniquesto integrate.
Whenwe are facedwith integralsthat involveradicalsof the forms and
we may makesubstitutionsthat involvetrigonometricfunctionsto eliminatethe radical.For example,to
eliminatethe radicalin the expression
we can makethe substitution
(Note:\thetamustbe limitedto the rangeof the inversesine function.)
whichyields,
The reasonfor the restriction is to guaranteethat is a one-to-onefunctionon this
intervaland thus has an inverse.
The tablebelowlists the propertrigonometricsubstitutionsthat will enableus to integratefunctionswith
radicalexpressionsin the formsabove.
Expressionin Integrand Substitution IdentityNeeded