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Calculus: Overview of Trig Substitution Strategy


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About this Lesson

  • Type: Video Tutorial
  • Length: 5:18
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 57 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: Trigonometric Substitution Strategy (3 lessons, $5.94)

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 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
Trigonometric Substitution Strategy
An Overview of Trig Sub Strategy Page [1 of 1]
So when thinking about an integral that might be primed for a trigonometric substitution, we're looking at integrals that have roots of things that are either the sum or the difference of perfect squares. When you see that, you know that you're primed for potentially being able to construct a right triangle, where either the hypotenuse is that square root that you see or one of the legs will be the square root that you see. And then once you have the triangle, picking one of the small angles as a theta allows you to create a dictionary for converting stuff with x's and square roots and stuff to trigonometric functions involving theta. And that will lead to the trigonometric substitution.
Now, when I actually think about these questions and I actually am asked to do them, I actually use this method, the method of constructing the triangle, looking at it, finding this dictionary, this correspondence substitutions between the x's and the trigonometric functions of theta. But there's always sort of a general template that you can memorize and use that might help. So let me show you, in fact, sort of a general overview strategy.
So if you see, in your integral somewhere, a^2, so some number squared, that's just a number, minus an x^2, then you might want to let x = a sin . And then you might be in good shape to actually use the identity, the Pythagorean identity, in the form 1 - sin^2 = cos^2 . If you see something that actually is the sum of two squares, where the a again is some fixed number, then you might want to use the strategy x = a tan . And then remember the Pythagorean identity in these guys, which is 1 + tan^2 = sec^2 . So there's another possible strategy. And if you see actually x^2 - a^2, then the hypotenuse here would, I guess, be the x. And so a substitution of x = a sec is a good idea. And then, looking at this equation slightly differently, you'd see sec^2 - 1 = tan^2 , and that might be a useful identity.
So this chart sort of provides an overview of how the method would look in the different cases. So you can think about this chart, or better yet, really understand the method and always realize how to make the substitution work. So, for example, if we had this substitution right here, if we saw this thing here, let's just take a look at the a^2 + x^2, let's just look at that together. If we had a^2 + x^2, then what I encourage you to consider is a picture that looks like this, and realize that, since I'm adding here, that must be part of the hypotenuse, something to do with the hypotenuse. So if I put an x here and an a here, then this length would be . And if that's the case and I call this angle here, then you can now list, produce a dictionary of all sorts of substitutions that are possible. And an easy one is to notice that the tangent - let me write this in my tangent font color - tan = , so . And if you solve this for x, you would see that x = a tan . And so, that is the suggested substitution, in fact, in this particular case. When you see that kind of object, the suggestion is to use that. And I think that instead of maybe memorizing all the possible scenarios, you can easily see how to actually derive the relationships very clearly through a simple little right triangle, being very cautious and careful what constitutes the hypotenuse and what constitutes the legs. Always, it's great when you try to produce a triangle like this, if you adopt this strategy that I'm proposing, to make sure that, first of all, this really is a right triangle, by checking that the Pythagorean theorem does hold with the way you place down the lengths. If it holds, then you actually have created a nice dictionary that will allow you to convert from to x's and back and forth. And if you found some mistake in there, then you just have to fine tune how you put down those lengths or how you represent the lengths, in order to make sure that everything gels.
In either case, the trigonometric substitution really does follow, in its heart, by substitution the x's lengths of sides of right triangles, to trigonometric functions of angles of that same right triangle. Okay, I'll see you at the next lecture.

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