[AM_20161105ITFF] False Friends in Integration (Calculus)

Question

Introduction

          False friends are words in two languages that look/sound alike, but differ significantly in meaning.  Do you know that there are also false friends in mathematics?  Can you distinguish and explain the difference between the two integrals?

Solution

          The integrand on the left has the variable  x  as the base and the constant  e  as the index.  So we integrate it using the Power Law.

          By contrast, for the integrand on the right, the base  e  is a constant whereas the index is the variable  x.  Integrating  e  to the power of  x  is the eeeeeeeeeeeeeeeasiest.  You just get back the same thing, plus the arbitrary constant of course.

Remark

          Many students make the mistake of trying to apply the Power Law for the exponential.  As a learner of mathematics, one needs to cultivate the habit of being observant and paying attention to detail.  This is part of developing one’s identity and character which is important in life.

Suitable Levels

* GCE ‘O’ Level Additional Mathematics

* GCE ‘A’ Levels (revision)

* revision for IB Mathematics HL & SL (revision)

* Advanced Placement (AP) Calculus AB & BC

* University / College Calculus

* other syllabuses that involve integral calculus

* whoever is interested