- Differentiateboth sidesso
- Changethe originalintegralin to an integralin :
where and
- Integratewith respectto :
- Changethe answerbackto :
Whilethis methodof substitutionis a very powerfulmethodfor solvinga varietyof problems,we will find
that we sometimeswill needto modifythe methodslightlyto addressproblems,as in the followingexample.
Example1:
Computethe followingindefiniteintegral:
Solution:
We note that the derivative of is ; hence, the current problem is not of the form
But we noticethat the derivativeis off only by a constantof and we knowthat
constantsare easyto deal with whendifferentiatingand integrating.Hence
Let
Then
Then and we are readyto changethe originalintegralfrom to an integralin and integrate:
Changingbackto , we have