Math 2300 Section 005 – Calculus II – Fall 2010
Substitution Examples – August 23–24, 2010
Guess and Check & Mechanical Substitution Examples

Solution: If you guess that is an appropriate antiderivative, then you differentiate to via the chain rule. But, we're now off by a scalar factor, which is an easy fix. Just multiply your antiderivative by half. We end up with
and we know this is the answer because we can differentiate to .

Solution: Let so that . Then

Solution: Letting we obtain and so

Solution: Let , so that . Then we have and

Solution: Let , so that (by the chain rule) and . Then

Solution: Since , we set and we have . We then have
Challenging Substitution Examples

Solutions: If we make the substitution then we have . This doesn't seem to help get rid of the remaining term in the integral, but notice that since we can solve for and write . Thus, we find

Solution: This example isn't immediately obvious at all. (Congratulations to those of you who spotted the path to the solution in class!) If we rewrite the integral and use (and so ) then we have
Definite Substitution Examples

Solution: Making the substitution and we find (in this case we do not change the limits of integration – they stay in terms of )

Solution: Using , and , we have (here we change the limits of integration to be in terms of )