Problem

Integrate each of the given functions. $$\int \fr…

Problem 18 Easy Difficulty

Integrate each of the given functions.
$$\int_{4}^{5} \frac{\sqrt{x^{2}-16}}{x^{2}} d x$$

Answer

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Calculus 1 / AB

Basic Technical Mathematics with Calculus

Chapter 28

Methods of Integration

Section 8

Integration by Trigonometric Substitution

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Video Transcript

so has everyone today and I'll think of an interim. You're 5 to 4. There's a definite integral and we need to use trigonometry substitution. So I hope you guys remember the three basic laws of uh the number of substitution. The given question is it will go export by 16 which is in the form of the third form which is she will give you X. Is it a spectator? Our aid would be route of 16 which is four. So actually be for sex data and X would be mhm integration and differentiation of this which is or sack teacher santa the theater. You know you have your major um operations and we just need to plug them in. So personally, um I need to take care of the endpoints. We have taken access for sector to so set up. Peter would be me taking the points in the form of theater. So data would be sick and worse extra for and doing that would give you integration of usual. Taking for putting four in terms of in in place of X. And 500 place of S. They would get zero and second word hi back for so I don't X squared 2016. You can read that as X. As or sec take a square 1916 or for stock. Kita square center Ds will be far check. Peter stand theater teacher. And this uh just you can actually see that you can simplify this further. So something further would give you sick and nerves five by four. Now simplifying the equation inside the room. Uh He would um get 16 6 square theater and taking 16 come in a country that this as 16 common sick square tita minus one over 16. Okay spirit Tater center for six tita tita. The theater. No um you can take you can even simplify this further. Bye cutting down the vegetables. The 16 and four will cancel out. Can you stack the second, the second council about then there will be for outside which are fans about this far. So that will leave you finding you. But root off and six square theater minus one can be last dance great data. So transfer detail enter upon 60 to wintertime teacher due to tell. And now Xena. Sorry about that six. You were five by four. Well that will give you 10 square Tito upon spectator. This can be again written as zero um Zero stack and wars by four. Um Sick is one hour costs. So of course can be taken outside and you can cut the cause uh and get can into 10 captain john miller from the sentinel costs scientist to our class theater. Into second version of one of our cost. So it will come to the numerator and it will cancel out leaving event. Finally. Sex and worse. Five by four. Um dan teacher. Scientific. Oh what I usually say that should always put it in terms of sine or costs and you have done so we're going to do that now. I live here with sine squared data or crossfire data. I need to bring it in terms of one either side of us. Now we have caused many no matter so we need to convert the numerator into um In terms of class. So you guys should remember the um the econometric identity science where take out flash crash fire teacher at six of the one and working that into the equation. Check his heel and when minus crossfire kita over across data now you can split this up and um but during the to upper boundaries and no memories of it and the handwriting is not that far but you guys know what is already. So I'll give you one or quashed Tito awareness, cash data. No if you read it this as you can rewrite this as uh sick integration of spectator minus near to check and where's five by four integration of course data. No. Um uh you can. Yeah I know I shouldn't be showing you the individual sect theater because it's pretty long but I'm sort of going to show it to you and go so I'm going to make a section right here as a spectator. You you can find it in the textbook. I'm pretty sure because it's a fundamental integral and it's where it's quite long to evaluate but I'm still going to do it right here so please bear with me if you already know the integral for sector to you can skip this part. Um So I'm going to make section right here. So the integral of sex to to now to do this it's quite tricky. Um you need to multiply the numerator and denominator help failure have everything written everything you don't know. Um I'm sorry you can give us, you do write tv everywhere. So did you overcome everywhere. There's an integral sign that's according to do that. And I think I was correct. So now we need to multiply the numerator and denominator. Bye factors second keita first and tita upon check you do the best time to to the theater. No writing this opening the brackets will give you sick spare data us sec X. And spectator tanta over sec. Yeah commentator I mean last time people the best thing here did peter now interview is U substitution for this integral. So you take us sexy to peter and going back to blue select green. Do you would be sector to Dante to use the change room and the multiplying room? Yes. Except yeah, security to um And you can reject this as um so this committee you ready? Peter and D. P. W. B. One or sick to go Dante to last second potato. Okay. That was quite long. All right. Selves and partners left. Um so writing that down. I'll give you sec. It's crazy to plus respect tita translator on. Um, so I really need more space with us. I'm gonna it is this because it won't fit over here. This last step one that so I'm going to make another line here. So to this we need you. So the next step would be citing the whole equation down and replacing the defeat a with what you've got here. So if we sec square theater us thanks of oil over here giving us um you so you took you as a sector to that's uh 60 atlantica. So we need to yep some other playing that would be do you over. Um You get you over firstly. They should be to you here, forgot to mention check you over. Uh huh. Selected at anti Douglas expert peter six sita. And you have a sec very data and now simplifying that do you over here. Um He was good. And then you which is sec data first anteater I forgot to mention that the U. S. Sector the first anti sector. Mhm. Be the first anti job question. Um There are many mistakes in this part. I made a few mistakes. So you would be a spectator done theater as an executive plus 10 theater not sectarian and theater. And yeah, I always kept repeating sector and eventual sectors of the standard or somewhere else. We had this foot for you. But like I said, and as you can get it, so you is actually sick the first and Peter I got confused with the plus and the D because they look similar in my handwriting. Uh and I did it. This integral is right. I was just saying it. Um I was just saying sectarian 10 theater, but in my mind I knew it 64 plus 10 theater. So that's what you get at the end of it. So this is the answer the interior of sick peter and now putting it back into the equation you would get. So you would get. Andrew there you go sick in order five by four and then sick get up stand peter. Oh and as you are integrating it. Um when this. Okay. All right. So we already integrated. So this next step would remove the integral thing. So don't be just and then sick data dance theater blessed. Anti Sorry keep on saying selective advantage. Um Sorry about that. And um the interval for causes sign. Societe to and this is from zero to park. I don't need to right outside our except in the side you too sick and worse five by four. And when you put that in your calculator you get alan to for the first term and minus three verse five for the second. And that is your final answer. Well that was pretty long goes. I don't know that. So what you basically did first you go to the terms for X and theta and you kept simplifying the interview simplifying the integral. Unless until you've got integral which you could actually integrated and the integration of sector does quite long. So I would advise you to watch a video on it or audiotex and read it. I'm pretty sure it gives you the integral shows you the integral for sector to the theater. And yeah so you keep simplifying it until you get the interval which can actually integrate and then you plug in your endpoints and we change their endpoints because um we change the main variable which is taken from extra theater so you can change the endpoints so that you don't need to put your final answer back in terms of X. So you can just keep it into the delta if you change the upper and lower values. So I hope you guys understood this. Um And thanks for watching.

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Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus

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Problem

Integrate each of the given functions. $$\int \fr…

Problem 18 Easy Difficulty

Integrate each of the given functions.
$$\int_{4}^{5} \frac{\sqrt{x^{2}-16}}{x^{2}} d x$$

Answer

Related Courses

Basic Technical Mathematics with Calculus

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Watch More Solved Questions in Chapter 28

Video Transcript

Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus

Heather Z.

Samuel H.

Michael J.

Joseph L.

Precalculus Review - Intro

Functions on the Real Line - Overview

Basic Technical Mathematics with Calculus

Heather Z.

Samuel H.

Michael J.

Joseph L.

Precalculus Review - Intro

Functions on the Real Line - Overview

Allyn J. Washington, Richard S. EvansBasic Technical Mathematics with Calculus

Heather Z.

Samuel H.

Michael J.

Joseph L.

Precalculus Review - Intro

Functions on the Real Line - Overview

Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus