[AP_Calculus_IGSB] Integrating an Exponential with Square Root

Problem

Introduction

     Here is an integration problem that has no clues as to what to do with it.  Hmmm ... the integrand (3 to the power of square root something) does not look like it can be simplified.  [H09, H10]  How about a substitution?  [H12]  But what substitution?  Usually, we substitute the “ugliest” part of the integrand.  What constitutes the “ugliest” requires experience and observation.  In this case, the square root expression looks pretty nasty.  But how do we even integrate  3  to the power of something?

How to Integrate the Exponential

Solution

Comment

     In this problem, the original variable of integration is  x.  When doing substitutions, it is usually easier to make  x  the subject, and then replace the  “dx” with its equivalent.  After the substitution [H11], we integrate by parts and then substitute back to express everything in terms of  x.

Heuristics Used

H04. Look for pattern(s)        [look for the “ugliest” part]

H05. Work backwards

H09. Restate the problem in another way

H10. Simplify the problem

H11. Solve part of the problem

H12* Think of a related problem

Suitable Levels

* GCE ‘A’ Levels H2 Mathematics (challenge)

* IB Mathematics HL (challenge)

* Advanced Placement (AP) Calculus AB & BC

* University / College calculus

* other syllabuses that involve integration

* any precocious or independent learner who is interested