Calculus 2 Lecture 7.3: Integrals By Trigonometric Substitution

Calculus 2 Lecture 7.3: Integrals By Trigonometric Substitution

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[@darthboganis1356]

What can I look for? 1:00 > Pythagorean theorem works. The function can be one side. [3:30] The idea of trigonometric substitution 7:00 Example 11:00 - x = tan(theta), dx/dtheta = tan (theta) > dx = sec^2(theta) dtheta integral of sec^3(x): kzbin.info/www/bejne/bInPhINrbdpgn9U > convert the problem back into terms of x The skinny on how to set up trig sub 18:30 (good conceptual discussion) Examples 36:00 Another example 54:00 >> look for a square root in your problem If a definite integral problem, change your bounds to avoid having to change back into x terms, which requires revisiting the triangle Example 1:11:00 > trig identities 1:24:00 If it doesn't fit, change to sines and cosines 1:26:00 Consider the function as it stands now. What method works best? Look at all the techniques > In this case, don't use more identities... but appreciate that U-sub works here. Reminder for U:Sub: You want to turn the function into something on the integral table. picking cos as u would leave a sin behind, but differentiating sin would cancel out the cos i.e. replace it with du Final example 1:32:00 Process: Is it in the table? U-Substitution? IBP? Trig-Sub, does it have a square root? > if it doesn't fit 'something squared minus something squared, change that. In this example, complete the square [1:35:20]. This was the idea because the stuff under the square root resembles a quadratic/polynomial 1:40:30 substitute to get rid of 'x+1' 1:52:00 sec^4(x) is an even power >> use identities [1 + tan^2(x)] to cancel out sec^2 via derivative and substitution rules > note: half angle identities are power reduction formulas. > video for integral of cos^4(x): kzbin.info/www/bejne/hISkqI2Er8-YrpY Substituting back 1:56:00

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[@boldapples]

Great introduction video! It's great that he tries to explain these substitutions using the most basic ideas and building from the ground up from them. It's amazing how much time this saves later on down the road when thoroughly explain these concepts at the beginning. So many other instructors treat this like memorization, and hand you formulas.

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Understanding how to set up your triangle is way better than memorizing which identity you should use. I was worried about my upcoming midterm at first, now I feel much more prepared.

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[@pjk7685]

someone posted this comment that I think more people should see higher up: For those who just happened to watch this video (because they are struggling with trig substitution) and watched up to 12:36, the lecture he's talking about is Lecture 7.1 starting at 1:16:39

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[@alexsholotyuk3258]

For those who just happened to watch this video (because they are struggling with trig substitution) and watched up to 12:36, the lecture he's talking about is Lecture 7.1 starting at 1:07:53.👍

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ty BUT its at 7.1 1:16:39

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I watched this lecture and felt like I learnt a tonne. I give another KZbin teacher a go on the same topic and the only reason I could follow along was because of this video. The other video was just a plug and chug formula video there’s no way you could learn and remember much from it. On the other hand professor Leonards lecture was unbelievable, a very good intro to the theory and why it works followed by fully worked solutions with explanation of each step, perfect way to learn 👍

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[@jacobharris3002]

It's been a few years since I took calc 2. But this is the first time I feel like I fully understand this as a concept. I was able to get through the class perfectly fine just memorizing the formulas but after I completed the class I often found I had to keep looking up the substitutions every time I had to use this method again. I had a friend taking calculus at the same time that told me there was a better way where you can derive the formula's each time by starting with the right triangle. I always liked this approach and wanted to use it but I never fully understood it because I didn't know what to pick for my sides. I didn't realize that there was some ambiguity to it and you could draw two possible right triangles for each substitution. And nobody told me that the other substitutions that we never use, i.e. with cotangent, cosine and cosecant work.This explanation finally made it click for me. It's not that we have to draw the triangle in the way you draw them, you just pick the triangle that allows us to integrate something more convenient. For the first integral we could have just as easily integrated the csc^3 instead of sec^3 to get the answer, which I verified just now. But integrating secant is a little less cumbersome than integrating cosecant.

[@discosheets108]

So grateful for this video, thank you! You slowed down the process of setting up that pythagorean triangle in such a way that I am now able to easily integrate with trig. sub. where I was totally stuck before. Two very enthusiastic thumbs up!

[@qbtc]

This video is three lectures in one and probably was given over a MWF schedule which is one week so don't get discouraged if it takes you a while to go through.

[@johnclymo3668]

These students are very lucky if this topic was taught over a week, I get only 2 hours to teach Integration By Trigonometric Substitution , I think that Australia has a different Education System.

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I got tired of scrolling down trying to find anyone who may have spotted the mistake at the end of the video (2:07) the term on the top right should be cubed not squared. Still awesome teacher and best in the world.

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[@georgesadler7830]

Professor Leonard, thank you for an incredible and lengthy video/lecture on Integrals by Trigonometric Substitution in Calculus Two. Trigonometric Substitution is not difficult; however a basic understanding of old fashion Trigonometry is needed. I also encountered Trigonometric Integrals in Engineering Physics and Electromagnetics. This is an error free video on KZbin.

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Thank you so much Proffesor Leonard, you have really helped me under this part of using identities🙏🙏 I had no idea about them