give-the-proper-trigonometric-substitution-and-find-the-transformed-integral-but-do-not-integrate-18

Question

Give the proper trigonometric substitution and find the transformed integral, but do not integrate.
$$\int \sqrt{25+x^{2}} d x$$

Sheryl Ezze · Accepted Answer

So we have this in a girl 25 plus X squared square rooted times dx. And we have to do the appropriate substitution trig substitution so that we would be able to integrate this. So If we use X April two and I can see this 25 here and I know I&#x27;m going to be squaring something and I basically want to create a pythagorean identity and we&#x27;ll find that I know I have the identity of one plus. The tangent squared of an angle is equal to the C. Can&#x27;t squared of an angle. And I basically want to manipulate to get this type of identity to come up. So I want this coefficient to be five times the tangent update to. That&#x27;s what I would like my substitution to be. And then notice if I differentiate I would find that dx is equal to five times the C. Can&#x27;t squared of theta times D. Theta. So now I have my substitution is that I can make not only for X but also for dx. So let&#x27;s do that. So I have underneath the radical I have 25 plus. And then we&#x27;re gonna square X. So we&#x27;re going to end up having 25 tangent squared of theta. And then this dx we need to substitute and will substitute this in place times five. C. Can&#x27;t square to data times did data. Now we just have to clean it up and do some simplifying. So if I underneath this radical I&#x27;m going to pull out that 25 and I have one plus the tangent squared of theta. And that&#x27;s what I was trying to create. And I have that five C. Can&#x27;t I swear to data times do fado. Now I can square root this And the square root of the 25 becomes the five. So we have five times. And then this value becomes the square root of she can&#x27;t square to fada times five. So you can&#x27;t square two Theta times D. Theta almost there don&#x27;t step by step. Mhm. And then I can take this five and this five and multiply them together and I can pull the constant out in front. I&#x27;ll do that. So I had 25 loops And 25. How I wanted to be in black. There we go. So I have 25. Now when I square root this item right here that becomes C. Can of theta. Yeah but I already have this C can&#x27;t squared of theta. So I end up with C. Can&#x27;t cubed of state to times D. Theta. So I&#x27;ve done my appropriate substitution with a trig substitution. And that original integral is equal to that in a girl. And we&#x27;re done we don&#x27;t have to integrate it.

Amy Jiang · Answer

Okay, so we have the integral of the square root of X squared minus 25 over X, the X Let&#x27;s draw out or triangle. So we have explored minus 25. So we know that our hypothesis is that the ex one of our sides is gonna be five. And the other side would be the square root of X squared minus five. All right, 25. Okay. Now let&#x27;s let&#x27;s or x equal to five, seeking dinner. So then, are the X is that that&#x27;s by sequins, Data, tangents, data. You did it Didn&#x27;t make it. So if we have ness Mmm. Hey, Should see Dixie. Mmm. Okay, So let&#x27;s figure out what tangents of data is. And you know, Veda is fine, but co sign. So we have this group of X squared minus 25 over five. So we have five. Tensions of data is equal status, girl roots of X squared minus 25. So this is what we want over here. You know, X is equal to five seconds later. So it&#x27;s plugging what we know. Get the integral of five tangents. Data over five. She can&#x27;t get up. Times are the X, which is five sequins data. And instead he did up. Okay, One of our fires cancel. Are she can also cancels five engines squared data data. Okay, we know that, but Chandon score data. That&#x27;s the sequel to seek inns squared theta minus one D data. Okay, they&#x27;re looking to read this. Get five times tangents of, But it&#x27;s five. I see. Well, what is tangent of data? We said tagged inhibitor is this thing over here? So we have five cents a square root of X squared minus 25 over five minus five. Data and data is equal to Let&#x27;s see X. You don&#x27;t have to hear this. That&#x27;s X over five and speaking in verse and the politics feeling of that five times. Thanks. Inverse of X over five. Let&#x27;s see, kid. Oh, it is. She don&#x27;t have a five year. They put out the five. It&#x27;s like this. So the solution is actually score boots up X squared minus 25 over five minus. Oh, not over fives. Because if I pull in our five cancels minus five seconds, inverse of X over five plus

Robert Daugherty · Answer

Section 84 problems 16. So I&#x27;m dealing with the indefinite integral here where I use tricks substitution. So when I see something that looks like a number plus x squared, the trick substitution here is going to evolve the tangents of X equal to five tangent of data. Then D X is going to be five seeking squared of theta do you data and then X squared plus 25 squared is going to be 25. Attention squared of theta plus 25 squared, which is going to be what 25 one plus tangent squared is just going to be seeking squared. So all of this squared. So this is gonna be five to the fourth, seeking to the fourth data so this integral transforms into X squared is going to be 25. Change in squared of fada, divided by this X square plus 25 squared is what we have right here. This is five to the fourth, seeking to the fourth of data. And then D X is equal to five Seacon Square. So five seeking squared of theta de theater. So it looks like and the u. N. I&#x27;ve got five to the third power The denominator. I have five to the first Power said this is going to be one fifth and then it looks like I have tension squared in the numerator and it looks like I&#x27;m going to have seeking squared in the denominator key. Hairless is right. Everything in terms of sign and co sign and see if there&#x27;s any cancellation. So the tangent that&#x27;s going to be the sine squared of theta divided by the coast sine squared of theta and the seeking is just gonna be won over the coastline squared of theta data. And what you see now is that there is a cancellation of the coastline in terms all go away. And so this just becomes 1/5 the integral of the sine squared of theta d theta. And our formula for this is this is 1/5 in a grove 1/2 one minus the co sign of tooth data D theta. And when I integrate this, I&#x27;m going to get 1/10 and that you will get Fada. We integrate the co sign, you get signs o minus sign of tooth data over to, plus a constant of integration. So this is 1/10 they had a minus. The sign of tooth ada is just too scientific Coastline data over to plus a constant of integration. So in terms of theta, this is 1/10 they, uh, minus sign Data CO Santa, plus the constant of integration. Now, what was our original? Substitution X is equal to five tension of their so original substitution X is equal to five tangent of data. That means that X over five is equal to the changing of Fada. So it means two things. It means that the Arc Tangent X over five is equal to theta If you draw a triangle. If the tangent of this angle is X over five, then the pot news has to be X squared plus 25 square root. So now I get my final answer in terms of X. So this is going to be 1/10 data is the arc tangent. So 1/10 our tinge it of X over five minus and then we just have to complete what is the sign in the co sign? So the sign of this angle is going to be X over the square root of X squared plus 25 and started me back up. There is a Let me just write it here. There&#x27;s up that 1/10 has to be there that&#x27;s distributed so sign of data and then the co sign of data has to be, what, five over the square root of X squared. I was 25 plus a constant of integration. So this, um, final answer becomes 1/10 arc tangent of X over five and that you see here the five in the 10 that become minus 1/2 and what you&#x27;re left with this just X over X squared plus 25 plus a constant of integration. So that&#x27;s our final answer. 1/10 our tended of X over five minus 1/2 X over X squared plus 25 plus a constant integration in the review here was making the trick substitution. And then they follow the same pattern, performing the integral integrating trick functions and then substituting back in for the the variable to get my final answer in terms of the original variable X

Pagadala Kishore Reddy · Answer

Okay, so we are going. Toby, You retaliating? The integral off D X way under hood off! Excess Queen minus. It&#x27;s queer By using them at it off. Trigonometry substitution. We have underwrote off extra square minus year script. So let X equals Yeah, seek and dig it up. That in place D X equals in tow seeking Dita Dan. Do that. And expert a year You close so you can&#x27;t get up? No, let us draw is a dangle bangle you be see second to Taik Wells Expert year. So yes, he&#x27;s thanks and BC ese Yeah, and a b e. He&#x27;s under route off excess square minus Yes. Quick. No integral off d explain Under Odo Extra square minus here. Square equals into girl off D x d axes ain&#x27;t music and Dada Dan Tito. So the exes in tow. So you can&#x27;t do that, Dan. Ditto Bye. Under hood off excess square. My next year square from triangle ABC. Yet Auntie Di calls one little dog extra square minus here. Script. So on the road of excess care minus C square equals into bandied about They&#x27;re equals integral off Yea gets canceled and dan did it and it gets cancelled. So integral Lab seeking to you, Don, did you don&#x27;t code equals integral off. Second reader Beaded off. You integrate. You&#x27;ll get Ellen off seeking Tita, place down. Do you go place? See? Good equals. Ellen. Off seeking. Tita seeking. Di di. Ease expert, you blows. Dante. Ada Dante. Days under a load off excess created minus years. Create way. Yeah. Bless. See that equals L N O X place on that rolled off excess queer minus C s square place. See? So in bigger love. D X bay under a load off excess square minus here. Squid equals Ellen off X Place Underwood off. Excess queer minus here square. Let&#x27;s see. Thank you.

Robert Daugherty · Answer

seven up. Four Problem 39 2 We&#x27;re dealing with an integral here that could be solved with trick substitution. Based on the format. Here, I know that the substitution is going to be five times attention of data. Then D X is going to be five seeking squared of data data and then X square plus 25 squared is going to be 25 one plus tended squared of theta squared. So one plus change it square dis seeking squared. So this ends up giving me what, five to the fourth and seeking to the fourth of data So this integral transforms into X squared is going to be 25. Change it squared of Fada, divided by five to the fourth, Seeking to the fourth of Dana Times D X, which is five secret squared of data. Do you think so? It looks like I&#x27;ve got five to the cubed over five to the fourth, so this is going to give me 1/5 and then it looks like I&#x27;m going to have tended squared of data divided by seeking squared of theta de data. So this is 1/5 um, tangent, iss sine squared overcoat sine squared of theta and then times Co sine squared off 8/1 D data. So this turns out to be just 1/5 the integral signs Great of data data. So I can write this as 1/5 integral 1/2 1 minus co sign to date A D data and I could integrate that. I end up with 1/10 and then that leaves me with fade up minus, you know, great. The co sign signed to data over to plus a constant of integration. Now, I use the fact that I know the sign of Tooth Ada. According to MIT, double angle formula is to signed a that goes on data. So that gives me this answer being 1/10 data minus Scient ada co signed data plus a constant of integration. So where are we now? What was my original substitution? It waas X equal five Attention of failure so x equal. Five tens of fada That tells me that the changing of fada is equal to X over five. So here&#x27;s my angle data. It has attention of X over five. That means that Sopot News has to be square root of X square plus 25. So this answer becomes 1/10 and then fade. A South Asia is going to be the arc Tangent of X over five minus The sign of data is X over square root of X Square plus 25 co sign is five over square root of X Square, plus 25 plus a constant. So this answer is 1 10 the art tension of X over five minus five X over X Square plus 25 plus a constant of integration. So that should be my final answer. Let me just double check that and that it&#x27;s all good.

Robert Daugherty · Answer

eight on for problem number seven tricks, substitution intervals. So I&#x27;ve got a definite integral 0 to 5 halves of one over the square root of 25 minutes x squared for into girls of this form. The substitution from a trig and metric standpoint is going to be X equal. Five terms, a sign of failure. This tells me that D X is going to be five co sign Data de theater and then the square root of 25 minus X squared is going to be the square root of 25 minus 25 sine squared favor. So this is going to be five times the square root of one minus sine squared data, which is five square root of co sine squared data or five co sign data. So I&#x27;m gonna use all of these pieces of information to transform this interval. Now, when X is equal to zero, that tells me that zero is equal to five sine theta or theta is he quit to zero. When ex secret of five halves when X is equal to five halves, that tells me that five halves equals five sine theta. Which tells me Scient Ada is 1/2. That means that the data is going to be pi over six. So this Inderal can be written as the interval from zero two pi over six. And then you write everything you have won over could be five coastline data. Five goes I&#x27;m data and then d X is five co signed data data. So this is just the integral from zero to power for 60 data, which is just data bad, awaited from zero Piper six, which is five or six. So the key here was recognizing the in a grand had this trick in a metric substitution. Then you have to work out. I mean, some of the bulk of the work is what&#x27;s the derivative? What&#x27;s the other value of the piece of the in a grand? Putting it all back together than the integration is usually pretty straightforward. At that point,

Problem

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

Problem 8 Easy Difficulty

Give the proper trigonometric substitution and find the transformed integral, but do not integrate.
$$\int \sqrt{25+x^{2}} d x$$

Answer

$x=5 \tan \theta, 25 \int \sec ^{3} \theta d \theta$

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Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus

Anna Marie V.

Kayleah T.

Caleb E.

Kristen K.

Precalculus Review - Intro

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$x=5 \tan \theta, 25 \int \sec ^{3} \theta d \theta$

Basic Technical Mathematics with Calculus

Anna Marie V.

Kayleah T.

Caleb E.

Kristen K.

Precalculus Review - Intro

Functions on the Real Line - Overview

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

Anna Marie V.

Kayleah T.

Caleb E.

Kristen K.

Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus