This page introduces us to Integration by Partial Fractions. This process often results in a situation requiring the natural logarithm and hence, some knowledge of its inverse, the Euler constant. That being the case, a brief overview of those concepts would be appropriate at this time. In the links directly below, James Tanton first illustrates Euler's constant (e), relating it to the compound interest formula. This is followed by the derivation of Euler's Formula by means other than the Taylor Series; although this is not a requirement at the high school level, it is nonetheless very enlightening and should be viewed by all. Beneath Tanton's offerings are two additional links, one providing an overview of the natural logarithm; the final link directly below provides a description of Integration by Partial Fractions. A small investment of time spent in each of these will serve us well as we proceed on through this process.
Euler's Constant "e"
Deriving Euler's Formula
Integration by Partial Fractions
The link below contains 23 pages of hand-written notes describing the Integration of Rational Functions. The focus of these notes is centered on the decomposition of rational functions into the sum of partial fractions; all but one of the examples chosen require the introduction of the natural logarithm to the integration process. For this reason, the first several pages in this set of notes focus on that function and its inverse, y = e^x. The example shown on pages 20 and 21 does not require the natural logarithm; it is integrated by Trigonometric substitution instead, a process that will be explained at a later date. For now we will work towards the mastery of Integration by Partial Fractions.
Integration by Partial Fractions - Samuelson
Visit my Introduction to Integral Calculus to view the entire process of integration as it progresses from the Power Rule, Integration by Parts and finally, the method of Partial Fractions. The process of Integration by Trigonometric Substitution will be added to this in the near future.
My Belief Statement
My Belief Statement: _It is better to be defeated on principle than to win on lies._ Arthur Calwell
Many students have a serious phobia when it comes to Math class, let alone writing an exam in that subject. I believe this phobia can be dispelled by "digging" into the "WHY " and the "HOW". In fact, many students WANT this. I attempt, wherever possible, to connect a geometric representation to its algebraic counterpart; an analysis of how the two versions interact is important. Contrary to what SOME individuals may think, I'm not at all afraid of technology. I AM, however, cautious to not "jump into the deep end"; we need to grow with this and "dovetail" the old with the new so that they COMPLIMENT each other.
I have several links in the sidebar at the right, as well as some of my own discriptions on various topics. These will grow in number and evolve over time, as will my hand-written notes; these can be viewed by following appropriate links which you will find throughout this blog. These notes will also be posted on my facebook site if that happens to be your preferred mode of viewing; I've included a link to that in the sidebar as well.