CK12 Calculus - Single Variable

2. Differentiateboth sidesso


3. Changethe originalintegralin to an integralin :

where and

4. Integratewith respectto :


5. 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