integrate-each-of-the-given-functions-int_04-frac9-d-tleftt29right3-2-3

Question

Integrate each of the given functions.
$$\int_{0}^{4} \frac{9 d t}{\left(t^{2}+9\right)^{3 / 2}}$$

Devanshu Khadka · Accepted Answer

So hello, have we gone today? We&#x27;re gonna solve the integral. Comes here to throw it nine over T square plus nine. The whole listed three by two. Now this is a econometric substitution question. And so t would be if you notice this in the form of X square. Yes things clear the whole room and you get the Can you use the substitution of excessive from a 10 ft on this? So um X as they come to a 10 m. Mhm. In this equation he would be able to three and peter because the are Square root of nine is 3. So duty moving on. The UK would be the differentiation of this. But you can do three sex. Where do you draw now? Um Okay duty here. Uh So now I mean you just need to replace these equations back into the main problem. So that leaves 02. Or actually now you you can change your endpoints because you&#x27;re going to take a reform theater. So they come from 02 People can be written as down and worse, three by three. So then it was zero is zero. And then this can be organized diamond wires. Sorry for the background knives. Yes. Women that Than was four x 3. Now writing this down, they can take the nine outside and this will give you one hour P square which is um 3 10%. 3 square is nine. So you can square theater plus nine for Asia three x 2 into DT which is trees. Sec square theater. Now um the in you can simplify this further, giving 02 and in worse four by three. Now You can take the nine common and you&#x27;ll get one where brian Coleman one work and nine times square Utah plus one. A recently by two. I have reckoned trail here. That&#x27;s that was my mistake. Mr three x 2 into into three fact square meter. No Uh transcript Pita Plus one gives you sex square theater. So do you go 10 and worse of four x 3. I need to read this in a better there. Yeah. Coming into my equation four by three. So now you can write this as 1/9. Okay. Paracetamol agent. Uh hft righto By two and 2 three. Check square meter. Um Now if you simplify this further, you would get uh, you can take the nine outside And get nine into 1/9. 02 integral from 0 to 10. And where&#x27;s four by three. Um, one over sick peter peter. So what did you do that river duty to don&#x27;t forget to write devi device. Um, Now, now that now you can, that&#x27;s quite a, is the integral. So you can some feathers a bit further by writing. It has The nine Guns Cloud Time and worst of four x 3 integral. From 0 to 9 and verse. Four by three And now six computers. Seconds. 2nd data is one of our cars data and this is one of our sector. As a cost. People will come on the numerator side and this will just give you cross peter peter. The integral. Of course tita is signed to um cell phone. So from that will be signed tita from its usual too. Mark and four by three. Part time for Betty. And now If you put this in the calculator, you get the answer. S four x 5. So I hope you guys understood that. That&#x27;s your final answer and thanks for watching.

Ernest Castorena · Answer

so he needs to be a partial fraction. See factor in T minus bust thing. It&#x27;s a team ministry marvel Were any dissatisfied e a times two plus thing bus being times a month, our A plus Pete visible to zero. So one of them&#x27;s the additive inverse of the other. And then three My minutes three b is equal to one. Well, we couldn&#x27;t supply this equation by three and that it so the bottom until lemonade beat. Throwing up with six n is equal to R is equal to 1/6. So be easy, Teoh. Negative one of this now could integrate those factors instead looking blood 1/6 Sylvia Ovary T minus two, minus one over t busting. And then we end up with 1/6. That scene We&#x27;re gonna need to take the natural log book and then subtract another natural law by not roll on properties. See Lanus being over teeth busting, plus an arbitrary constant. Okay,

Tom Greenwood · Answer

Okay, here we have our integral on. The first thing we want to do is take this Fraction one over T squared minus nine, and we want to decompose it using partial fractions. Recognize that this factors into a T minus three and a T plus three, which will be the denominators of our fractions. So we would have a over T minus three plus B over T plus three being the form of the decomposition. Multiply everything by our common denominator. That way we just have one on the left hand side on the right hand side, each of these factors will reduce and we&#x27;ll have our numerator times the other denominator. So that would give us a Times T plus three a. And then be times the T minus three. So that would be b Times t mine&#x27;s three B. Now that can equate the terms on each side. We don&#x27;t have any t term here and we have a plus B. T. So that means that a plus B has to equal zero and we equate are constants. So the three a minus the three b past equal one. So solve this by whatever message you want to it by hand. Or grab calculator and use the matrix operations. What have you we end up with? Hey, peopling 1/6 and be equaling negative 16 So not the prettiest of values here for ah, for A and B gotta What was what we have? So we have the integral From 0 to 1 of now we can rewrite our fraction as our decomposition of 1/6 over the T minus three minus 1/6 over the T plus three times are DT. So now I can factor out the 1/6 and then I can also separate these because of the subtraction into two different tentacles. So we&#x27;re gonna have 1/6 times the integral from 01 of one over T minus three DT minus the integral from 0 to 1 of one over t plus three t t. So now we can use our one over u times d. You ruled from the integration rules that we have let u equal each denominator and do you is just the DT. So we don&#x27;t have to do any kind of substitution or anything. So we have 1/6 times the natural algorithm of absolute value T minus three minus the natural algorithm of the absolute value of T plus three. And that&#x27;s gotta get evaluated from 0 to 1. I&#x27;m gonna combine these into one algorithm first. That&#x27;s 16 times in natural log, uh, T minus three over t plus three. And that&#x27;s evaluated from 01 So we have 1/6 times evaluating these. Substitute one in here. We&#x27;ve got the natural. All the rhythm of the absolute value one minus three over one plus three minus the natural algorithm of the absolute value. Uh, zero minus three over zero Paul story. Okay, well, that&#x27;s gonna be negative to over four. So that a bidding algorithm of 1/2 and that will be negative. Three over positive three, which is negative on absolute value, that it&#x27;s positive one. And the algorithm of one is zero. So this is just one sex times the natural over them of 1/2

Aishwarya Krishnakumar · Answer

So for problem 29. Where assets? All of this definite intergroup. And we&#x27;re going to use u substitution since we have a function inside for functions. So we&#x27;re gonna let you equal three plus eat to the two teams. And if you take the derivative of both sides, you would get two need to the to D Katie and we&#x27;re gonna go ahead and move this to to the other side sober. We have 1/2 do you and then the two goes away. And as you can see, we have a this We have it right up here in the original. You couldn&#x27;t. So that&#x27;s gonna help us when we write the actual integral. But before we do that, we have to change the bounds. Since these bounds are related to X and we have to do a u substitution. So we have to change these bounds. So that is related to you. So to do that, we just have to plug in both of the bounds. Oh, my bad. These air actually related to t not ex. There are no exes in this problem. A little zero because t and one he calls and we&#x27;re gonna change those teas into use. And to do that, you have to plug it into this equation. So we&#x27;re gonna say three plus e to the two times one. Since we&#x27;re plugging in the upper bound into this tea and then to find a lower band, we do three waas e to the two times zero. So if we do that, we get the lower bound to be four in the upper bound to be three plus knee squared, and then we can go ahead and do the actual u substitution so it would be won over. We plugged with you into the original equation and we will get you squared. And then for the green circles, that would be 1/2. Do you 1/2 to you so we can move the 1/2 to the front sense of the constant. We have the same in a girl. And to make things easier, you can also move the U to this second power to the numerator. So it would be you to the negative to power. And now all you have to do is take the undergrowth until you add one and there would be you to the minus one times a minus sign. And don&#x27;t forget this 1/2 and you would find this with respect to four and three plus U squared. So multiply the negatives, turn out and then it would be one over you plug in three plus c squared into you will be three close e squared And what I did was moved the move you to the denominator and then minus won over Plug in the for into the U And if you plug this into a calculator, you get zero point zero seven. I think eight.

Xiaomeng Zhang · Answer

we all know that the relative off each other power off to T plus three just equals two times. But we&#x27;re sorry to times I need to the power off to key. So we are uses up a Cugini in those question So sorry blocks e to the power of two key Squire and the the relative off beat the power of two people. Three Here we need to comes one tryst here because it&#x27;s too here and those are We need to change off our range status he to the power off to He loves three need the power off kooky plus three It&#x27;s a new indie grow You call to you to the problem of to a plough straight those wankers before run over Oh, over you, Squire You So those when you go to u to the power off miners Well, over minus one my nose. One christ The ghost approximately e counts. You pawned zero horned 0768

Matt Just · Answer

Okay, So another definite integral from negative to the negative one of this function. Negative to over tea plus three e to the zero point three team Dean. So anti derivative. So? Well, just remember that. Um okay, so, So anti derivative of one of her t is log absolute value of tea, since it&#x27;s going to be negative too. Log Absolutely. T plus. Okay, well, this is going to be three eating a zero point three t, but I might have to divide by this constant factor. I just think when you take the derivative, you multiplied by zero point three by the chain roll you were just going to divide. So I have three divided by zero point three. Me too. The zero point three t evaluated from negative to donate one. Okay. So again, we can just actually evaluate each one of these turns at the end points and take the difference. So and you have two times log to minus log one and notice that I&#x27;m applying the absolute value to negative one and and negative, too, and actually ended their records. So long. Absolute value of one minus long. Absolute values too. Plus Okay, three overs your point three. That&#x27;s one over&#x27;s your point. One or one over one tenth. Let&#x27;s just ten. And then we have into the minus zero point three. Brian&#x27;s A to the minus, sirrah point six. Okay? And so that&#x27;s good. I mean, one thing we can do. Okay, Log of one zero. So this is just to sow the negatives. Cancel natural, too. Plus ten times this difference. At this point, we can probably just plug this end and what we get Approximately three point three zero six.

Problem

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

Problem 10 Easy Difficulty

Integrate each of the given functions.
$$\int_{0}^{4} \frac{9 d t}{\left(t^{2}+9\right)^{3 / 2}}$$

Answer

$\frac{4}{5}$

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Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus

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Catherine R.

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Precalculus Review - Intro

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$\frac{4}{5}$

Basic Technical Mathematics with Calculus

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Catherine R.

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Precalculus Review - Intro

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

Grace H.

Catherine R.

Kristen K.

Samuel H.

Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus