identify-u-d-u-and-the-formula-from-appendix-d-that-would-be-used-to-complete-the-integration-do-n-2

Question

Identify $u, d u,$ and the formula from Appendix $D$ that would be used to complete the integration. Do not integrate.
$$\int \frac{x d x}{\sqrt{x^{4}-16}}$$

Vinnu M · Accepted Answer

so here in this question I a few more media pill you as X squared. So do excuse me Deal by two as extra dx No. Also the put on the Lord limit off you would be cereal and four respectively. No, Using this information, I can rate I, as in typical off square wrote off 16 minus youth choir Do you and have outside but delivered Corson zero to for soldering this I have eyes equal toe when by two multiplied by you buy to plant by 16 minus you squared plus 16 by two signing worth off you by four but a limit goes on zero four So after soldering this I have have multiplied by it multiplied by bye bye to so after soldering days I have I s to pay which is

Geena Pullo · Answer

Hi. So, in this question, we&#x27;re going to be solving for the indefinite integral using the method of you substitution. So we have the integral of X over the square root of one minus for a woman&#x27;s X to the four and then times rt acts. So the first thing we&#x27;re going to do is we&#x27;re going to just manipulate this equation a bit. So we&#x27;re going to have the integral of X over the square root of one squared minus X squared, squared D X. And now from here we&#x27;re going Teoh set Are you equal to X squared? And then we&#x27;re gonna just take the derivative of both sides so we&#x27;re gonna have to you is equal to two x d x and d you over two acts is equal to D acts, and all we&#x27;re gonna do from here now is replace our x squared with Are you and r d X with R D you over to X. And so we&#x27;re going to have the integral of X over the square root of one squared minus. You squared times d you over two x and we can cross out our exes on. We can rewrite this as the d u did a girl of D you over one squared minus you squared times one half. And so now we&#x27;re going to do is use a formula that says that the integral of D you over the square root of a squared minus the, um, minus X squared. Sorry, expert. You squared this case, um, is equal to the ark sign, Have you over a plus. See? So now we&#x27;re just gonna rewrite our equation like that. So we&#x27;re going to have one half times the arc sine, uh, you because it&#x27;s you over A and R H is equal to one start. All right, that right here are a is equal to one and are you is equal to you for right now. So we&#x27;re going to just have are you plus see? And now from here, all we have to do is we have to rewrite this as, um are u is equal to our X squared. So we&#x27;ve to just plug that in. So we&#x27;re going tohave one half times the or we could just say, actually, um, the Ark sign of X squared over two plus c and that is going to be our final answer.

Julian Gerber · Answer

So here we have the integral of X over the square root of one minus X to the fourth DX. Um, And to solve this, we&#x27;re going to use a trig substitution. So instead of acts, we can set X equal to the square root of sign you. And the reason we&#x27;re doing this is because ultimately we&#x27;re going to want to use the trick. Identity one minus, um, signs squared of you are X equals ko sine squared of you. Um and we see we already have in the denominator, you&#x27;re one minus X to the fourth. So you&#x27;re starting to see how we&#x27;re going to get this trick identity here. So if X is root, sign you than DX de If we differentiate d x over d you, we get one half sign you, um, one half sign you to the negative one half times co sign of you by using the chain rule. Um, and the power rule do you right, cause we want this in terms of you and now we can just substitute. So instead of X and the numerator, we have square root sign you the denominator. We have square root of one minus um X to the fourth, which becomes square. Sign you to the fourth. Which, if sign if the square root of sign you is the same thing is writing sign you to the one half. Then this just becomes sine squared. You. So we see how we&#x27;re going to use this trick identity here. And then instead of dx we substitute and one half sign you Negative one half co sign you. Do you? Um, now there are a couple of things we can do to simplify. Now, first of all, we can take the one half here outside the integral because it&#x27;s constant. Next this sign you to the negative one half because its a negative power that is the same as one. Same as one over square root sign you. Which means that this cancels out with this, um, the square root sign you in the numerator. Um, So all we&#x27;re left with is, um, co sign you, do you over square root. And then we can use this trick. Identity one minus sine squared. U equals co sine squared you. We can use that to just substitute in co sine squared you here. Now we&#x27;re starting to see how this might get Very simple. So square root of co sine squared is just gonna be koko sign. So this becomes one half times the injured girl co sign you over. Co sign you d you these two co signs, of course. Cancel out. And all we&#x27;re left with is one half times the integral do you, which we know from the integral table is just one half you, plus the constant of integration. See, now all we have to do is substitue back in for X. So in order to do that, we&#x27;re going to use the original equation here. We used, um X equals square sign you. If we square both sides, we get X squared equals sign you, which means that you equals the inverse sign of X squared. Now, using this substitution, um, we can just plug, um, this back into the answer we got here. So we get one half times the inverse sign at squared, plus the constant of integration

Ashley Boni · Answer

instead of the integral and, uh, figure out which role we would use on page. I believe it&#x27;s 5 16 so we want to look at the end of roll one over X. I&#x27;m square root of X squared minus board E X, though this looks exactly like Rule 20 with with You equal to X and a sequel, which is the inverse Citigroup Room.

Margaret Keymakh · Answer

Okay, So for this problem, we&#x27;re gonna be using integration by parts to find this into role right here. Um and the problem ready tells us what you nd of you&#x27;re going to be. So in this case, you is equal to X, and Devi is equal to co sign of four acts. Okay, so now let&#x27;s find the other parts that we need to plug into the formula. First thing that we need is d&#x27;you. So we take the derivative both sides here do you is just gonna be equal to DX. We also need to know what just to be a So we have Devi. We have the derivative of V, but we don&#x27;t have just be on its own. So we need us. Do we need to go backwards and take the integral V is going to be equal to the integral of the co sign before axe. And to do this, you can use your substitution. But is it pretty easy once we can do in our head, we&#x27;re just gonna have a 1/4 in front here, uh, and integral of co sinus signed. So he&#x27;s gonna leave off. Please leave us with sign of for X and guest is an indefinite integral. So we should have a seat here. But we&#x27;re gonna add that bend. So we&#x27;re not gonna worry about it right now. Okay, so now we have all the parts that we need to plug into the formula. So let&#x27;s do that. This integral is gonna be equal to you, Tim Z. So in this case, you is X and V is 1/4 sign of four X. So we&#x27;ll apply by that, um we&#x27;re gonna subtract the into girl of sign four X over four plugging and for V and times D excuse to use equal ddx. Okay, so now let&#x27;s simplify this here. This part, we just rewrite as X sign for X Divided by four minus on again. This is a pretty simple U substitution this for here could just be treated as a 1/4 constant in front of the integral of this is going to give us co sign of four ax. It would be over four, but times 1/4 is gonna give us 16. So over 16 and now we&#x27;re gonna add the plus C like I mentioned before. Since this is an indefinite integral and so they were done. This is our final answer

Problem

Identify $u, d u,$ and the formula from Appendix …

Problem 4 Easy Difficulty

Identify $u, d u,$ and the formula from Appendix $D$ that would be used to complete the integration. Do not integrate.
$$\int \frac{x d x}{\sqrt{x^{4}-16}}$$

Answer

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Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus

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Basic Technical Mathematics with Calculus

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

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Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus