**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**

*x*. When doing substitutions, it is usually easier to make

*x*the subject, and then replace the “d

*x*” with its equivalent. After the substitution [H11], we integrate by parts and then substitute back to express everything in terms of

*x*.

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

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