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Calculus: Integrals of Other Trig Functions

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About this Lesson

  • Type: Video Tutorial
  • Length: 9:02
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 96 MB
  • Posted: 11/18/2008

This lesson is part of the following series:

Calculus (279 lessons, $198.00)
Calculus Review (48 lessons, $95.04)
Calculus: Techniques of Integration (28 lessons, $40.59)
Calculus: Integrals - Other Trig Function Powers (3 lessons, $4.95)

In this lesson, you will learn how to integrate tangent, cotangent and secant. You will see how solving tangent and cotangent antidifferentiation problems will generally involve expressing them in terms of sine and cosine and then applying u-substitution to the problem. Integrating secant and cosecant, on the other hand, involves multiplication by a specific fraction that is equal to one. Professor Burger will walk you through what the integrals are of these trigonometric identities as well as how one would arrive at them.

Taught by Professor Edward Burger, this lesson was selected from a broader, comprehensive course, College Algebra. 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'Hôpital'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.

About this Author

Thinkwell
Thinkwell
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Founded in 1997, Thinkwell has succeeded in creating "next-generation" textbooks that help students learn and teachers teach. Capitalizing on the power of new technology, Thinkwell products prepare students more effectively for their coursework than any printed textbook can. Thinkwell has assembled a group of talented industry professionals who have shaped the company into the leading provider of technology-based textbooks. For more information about Thinkwell, please visit www.thinkwell.com or visit Thinkwell's Video Lesson Store at http://thinkwell.mindbites.com/.

Thinkwell lessons feature a star-studded cast of outstanding university professors: Edward Burger (Pre-Algebra through...

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Recent Reviews

Nopic_tan
Awesome examples!
07/20/2009
~ brittanie

Great use of examples to explain how to find the derivative of a trig function. I hate taking the derivative of sec(x) and really appreciate this video for helping me learn how to live with it. :)

Nopic_tan
Awesome examples!
07/20/2009
~ brittanie

Great use of examples to explain how to find the derivative of a trig function. I hate taking the derivative of sec(x) and really appreciate this video for helping me learn how to live with it. :)

Techniques of Integration
Integrals Involving Powers of Other Trig Functions
Integrals of Other Trigonometric Functions Page [1 of 2]
All right, sines and cosines are one thing, but what about tangents and secants? Can we manipulate those? Now, the answer, by the way, if you’re saying, “Gee, how come we’re just seeing kind of a hodge-podge of trig functions being pushed together under one integral sign? Shouldn’t we give an opportunity for everyone to cohabitate with everyone else?” The answer is if you come up with your own individual combinations, quite often those functions can’t be integrated. They're just too hard. So we’re looking at those functions that actually can systematically be resolved using certain techniques. So I don’t want you to think these things are being pulled out of thin air. The reality is they are being pulled out of thin air; I just don’t want you to think that. The reality is these are particular types of combinations where we can actually say something and actually answer to question of what’s the integral.
Okay, so I want to warm up to looking at tangents and secants by just doing a really fast review of just what’s the integral of tangent. So what’s the integral of tan x dx? When you think about it for a second, you go, “Gee, now what function is it whose derivative is tangent?” Answer, I don’t know. So let’s not do that. In fact, when I see tangent, I usually think about sines and cosines, since those are the only two trig functions that I know anything about. So let me immediately convert this to sines and cosines, which I can do by remembering that tangent is just . Now once I do that, I now feel really good, because I see this could make a very nice u du-substitution. And the u du-substitution would be to let u equal the bottom. And notice that the derivative of the bottom is almost sitting on the top. So let’s make a little u-substitution. Sub people call this, by the way, a u-sub. Let’s make a u-sub, but I’m not hip, or new or aged, so let’s just make a u-substitution. Let’s let u = cos x. If I do that, if I differentiate, I see that du = -sin x. And if I multiply through by that negative, it will pop up on this side. And now, let’s make a substitution. What about this piece right here? Well, that’s in the underground, you notice. That’s sort of underground. So if we do the underground thing, that would be just u. And what about the overground? Well, I’m left with a sin x dx. And that’s precisely –du. So, in fact that’s precisely negative, I’ll pull the negative sign out, du. Well, that’s actually a pretty easy integral to look at, because that’s just negative, and then this is just the . And what’s u? Well, let’s now reduce the shorthand and put in a longhand. If you put in a longhand, you see . Now, that’s perfectly fine and, in fact, that’s great and that’s correct. Some people don’t like the negative sign. Let me just show you, in case you might have heard a different answer to this question – it’s actually the same answer written incognito. What I could do is remember that a coefficient in front of the natural log can be made into an exponent. And if I use that little property of logs, I could write this as . And what is that? Well, that’s just the natural log of – well, the –1 power means . And what is ? That actually has a name, that’s secant. So this would be the . And that might be a more standard way of saying with the integral is. So, in fact, this integral is a little bit weird. The integral of tangent is not just one you can roll off, like the integral of cosine. You can say, “Oh, it’s sine.” The integral of tangent turns out to involve a couple of steps and the answer is . Using the exact same ideas, just a different substitution, you can actually verify for yourself that the integral of cot x is actually the – well, you write it a whole bunch of different ways. Let’s write it this way, . A similar-looking thing. So, in fact, you can find now the integrals of these kind of guys.
But what about the integral of secant? Now, that one is a real pill. So I want to show you this one and I want to just give you a sense of how you can find this one. This is a tricky one. Now, I’ll tell you what the answer is. I'm just going to jump right to the chase here and tell you what the answer is. The answer to this is . Isn’t that awful-looking? It sure is awful looking to me. Where does that come from? It came from a little teeny trick, and I'm going to set up the trick for you and then I’ll let you try to try it on your own. And the trick is this – you know, when a problem looks really hard, a lot of times it’s a really good idea to multiply by the number 1. If you multiply by the number 1, you don’t change anything, but sometimes you can make the question a little easier to tackle. And in this case, what you want to do is you want to multiply sec x by 1, and the very clever choice of 1 you’re going to multiply sec x by is the following: sec x + tan x. And then we have to put it on the bottom, too, so we don’t change the value. It’s just the number 1. So take the sec x and insert right there, right in between there put in this term. Put in . This is a trick, not at all obvious why we’d ever want to do this in life, but stick it in right there. So now you’re looking at this. You can make a u-substitution and the u-substitution should be this: let u equal the bottom; sec x + tan x. And if you let u equal the bottom, the what happens? Well, what's du? Well, du equals – but what’s the derivative of secant? The derivative of secant is actually sec x tan x plus – and what’s the derivative of tangent? That’s sec2 dx. But look at what you have on the top here. You have a sec x sec x and a sec x tan x. This entire top is nothing more than du. So when you make that substitution after this little clever trick, what you actually see is that our integral becomes the following: . And that integral is easy, that just equals . And what is u? Well, u is that awful-looking thing right here, so . And that’s what I had announced earlier. So, in fact, if you multiply it by a very clever choice of 1 and then make a -substitution, you can actually even find the integral of secant. So that’s great.
Now, what I want to think about is what happens if you start to have powers of tangent and powers of secant and so forth. You think about that and we’ll talk about it together up next. I’ll see you there.

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