rulesis neededto solvean integrationproblem.
LearningObjectives
A studentwill be able to:
- Computeby handthe integralsof a wide varietyof functionsby usingthe techniqueof -substitution.
- Applythe -substitutiontechniqueto definiteintegrals.
- Applythe -substitutiontechniqueto trig functions.
Probablyone of the mostpowerfultechniquesof integrationisintegrationby substitution.In this technique,
you choosepart of the integrandto be equalto a variablewe will call and then writethe entireintegrand
in termsof The difficultyof the techniqueis decidingwhichterm in the integrandwill be best for substitution
by However, with practice,you will developa skill for choosingthe right term.
Recallfrom Chapter2 that if is a differentiablefunctionof and if is a real numberand
then the ChainRule tells us that
The reverseof this formulais the integrationformula,
Sometimesit is not easyto integratedirectly. For example,look at this integral:
.
One way to integrateis to first expandthe integrandand then integrateterm by term.
That is easyenough.However, whatif the integralwas