Integration by Parts
Let us reflect back on the process of differentiation for a few moments. This entire process began with our wish to find an expression representing instantaneous velocity at any given point on a function; we found that this value was represented by the slope of the tangent to the function at that point. The expression for this was discovered through first principles (delta method), and is now referred to as the derivative of the function; we used first principles to find derivatives of several different types of functions in order to more fully conceptualize the notion. As our functions became more complex, this process became much more arduous and time consuming so we were introduced to some helpful rules to assist us in our work; they are the power rule, the product/quotient rule, and the chain rule.
Integration is essentially the “inverse” process of differentiaion and has its own set of “rules” that can assist us in determining the area, among other things, under a curve in a given interval. We have already been introduced to the power rule for integration in my Introduction to Integral Calculus. The power rule works very well for relatively simple functions; as these functions become more complex the neccessity for other rules emerges, just as it did with differential calculus. The next “rule” that we will familiarize ourselves with is referred to as Integration by Parts, a process tied directly to the product rule of differentiation; the notes contained in the link below will illustrate this. In these notes, a comparison between integration using the power rule and integration by parts will be made early on as they relate to simple polynomial funtions. The purpose behind this is to help us become accustomed to this new “rule”; as we work through the notes, the functions being integrated will evolve into more complex ones, hopefully leaving us with a deeper appreciation for this new process.
Integration by Parts – Samuelson
For additional information on Integration by Parts, click on the links below. For easy reference, these links can also be found on my Math 31 Blog in the sidebar under Integral Calculus.
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