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Calculus: An Introduction to the Integral Table

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About this Lesson

  • Type: Video Tutorial
  • Length: 6:39
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 71 MB
  • Posted: 06/26/2009

This lesson is part of the following series:

Calculus (279 lessons, $198.00)
Calculus: Techniques of Integration (28 lessons, $40.59)
Calculus: Integration Using Tables (2 lessons, $2.97)

Taught by Professor Edward Burger, this lesson comes from a comprehensive Calculus course. This course and others are available from Thinkwell, Inc. The full course can be found at http://www.thinkwell.com/student/product/calculus. The full course covers limits, derivatives, implicit differentiation, integration or antidifferentiation, L'Hopital's Rule, functions and their inverses, improper integrals, integral calculus, differential calculus, sequences, series, differential equations, parametric equations, polar coordinates, vector calculus and a variety of other AP Calculus, College Calculus and Calculus II topics.

Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.

He has also taught at UT-Austin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the 2004 Chauvenet Prize, and the 2006 Lester R. Ford Award, all from the Mathematical Association of America. In 2006, Reader's Digest named him in the "100 Best of America".

Prof. Burger is the author of over 50 articles, videos, and books, including the trade book, "Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas" and of the textbook "The Heart of Mathematics: An Invitation to Effective Thinking". He also speaks frequently to professional and public audiences, referees professional journals, and publishes articles in leading math journals, including The "Journal of Number Theory" and "American Mathematical Monthly". His areas of specialty include number theory, Diophantine approximation, p-adic analysis, the geometry of numbers, and the theory of continued fractions.

Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.

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Techniques of Integration
Integration Using Tables
Introduction to the Integral Table Page [1 of 1]
Oh, my, it's dark in here. Get a little light. Let me open up some shades. Oh, there you are. Oh, welcome to calculus. Finding integrals is a really, really, challenging thing. In fact, it's a very challenging endeavor, because most functions, in fact, can't be integrated. And this is a sad truth that we should begin to believe now, that all integrals are not created equal. That, in fact, really, within the spectrum of integrals, there are those that can be integrated and those that can't. And those that can't are the vast majority. And those that can are the minority. So how does one go about finding the integrals for functions that we just don't know off the top of our heads how to integrate?
Well, one way to do this is to consider the table setting, which is a sort of formal thing. So if you take a table, for example, like this, this is a lovely table. In fact, you can even put on a candelabra, if you so desire. And on the table you can set this with all sorts of integral formulas. For example, you can put on formulas like this or like this. And these are a whole bunch of identities that really, really old people - well, they're now dead, maybe they weren't old when they found them - worked out for us to use and enjoy. And this is called the method of integration using a table. And you can look up whatever integral you wanted. This is quaint and lovely, how you can just use a little table to actually find an integral.
Let me show you the utility of this table with a particular example. So let' us now consider the following integration question. So let's consider the following integral. Let's integrate the . Now, that integral looks extremely threatening, and indeed it is. And yet, this is an example of an integral that we can find, but by actually using the table setting. So let us take a look and reexamine the table.
If we reexamine the table, you can see that one of these formulae actually corresponds quite attractively with the question at hand. In particular, this one seems to be a particular utility, since it has the general form of the question at hand. In particular, it's an integral of . So when I see that, I can actually use my table and the general answer of it, and actually figure out exactly what this integral is by merely copying the answer from here and inserted the particular value of a. In this case, since I see an a^2 here and 4 here, a must be 2, since 2^2 = 4. So, using the table, I can report the following: this equals . And you can check the answer by actually taking the derivative of this and seeing that, in fact, you get this.
So, welcome to the 15^th, 16^th and even 17^th Century of mathematics. And this works great. But you know what? Why bother? In fact, the reality is that we're no longer in the 18^th Century. In fact, now we're not even in the 20^th Century. In fact, I don't even know what century this is, because maybe you're watching this lecture and the year is 3014, which means that I am dead, and that's depressing. But in any case, the point is that we need not use these ridiculously arcane methods of looking things up. We need not use the flame of books and so forth when, in fact, we have the modern day computer. And with the computer, you can actually just type in a function and either spit out a numerical value if, in fact, you have a definite integral. It'll just tell you numerically what it is to any precision almost that you want. Or, if you prefer, there are even programs that will symbolically report things and you can actually report things of this form and see the answer symbolically.
So, the quaint notion of the table, in all its glory, and the quill and the candle are indeed just that - quaint. And the utility medium. I'll see you at the next lecture.

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