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MentWshtia: 61.2128Consider the definite integral7.(25 22)1/4 dx.What is the substitution to use? give the one that is the most efficient:) du For this correct choi...

Question

MentWshtia: 61.2128Consider the definite integral7.(25 22)1/4 dx.What is the substitution to use? give the one that is the most efficient:) du For this correct choice, dx(There can be more than one valid substitution;If we make this substitution, then the integral becomes of the formf(u) du. What are a b and f(u)?"76 = 37f(u) Finally; use this work to computeJo Tx (25 _ 22)1/4 dc Give the exact value:

ment Wshtia: 61.2128 Consider the definite integral 7.(25 22)1/4 dx. What is the substitution to use? give the one that is the most efficient:) du For this correct choice, dx (There can be more than one valid substitution; If we make this substitution, then the integral becomes of the form f(u) du. What are a b and f(u)? "7 6 = 37 f(u) Finally; use this work to compute Jo Tx (25 _ 22)1/4 dc Give the exact value:



Answers

Evaluate the integrals by making appropriate substitutions. $$\int t \sqrt{7 t^{2}+12} d t$$

In this problem we have given integral assigned to the power seven x dx on the objective is to evaluate the integral by using appropriate method. We can write this integral as integral, consigned to the power six eggs in to cosign X'd eggs It would be equals 20 Grill Who? Science square eggs Whole power que in to cosign x d s by using the pregnant metric identity that is science square eggs is equals toe one minus science square eggs We can write it as integral one minus sign square eggs whole Q into cosine X DX Now we will substitute sine X as equals to t differentiating both side We get design x d x is equal to d t So here we will substitute t for Synnex on d t four cosine x dx We get integral one minus t square. Who will queue, baby By using the jailbreak identity The tightest integral one minus 32 square plus three into T square. Who's where? Minus t square whose que d t it would be equals 20 girl one minus three T square plus three and two t to the power fall minus P to the Power six DT. We can write it as integral DT minus trained to Integrity's where DP Plus trained to integral t to the power before deity minus integrity to the power six BD. Now we will integrate each term one by one we get integration off PTSD minus three into integration off T square. DT is a cube upon tree plus trained to integration off day to the pub before deity is equals toe pay to the bubble five upon t minus The integrity to the power six would be equals toe t to the power seven upon seven Classy. It would be equals toe B minus. Take you plus three upon five into t to the power five minus. Stay to the power seven upon seven plus C Now we will replace sine x 40 on we get sine X minus Sign Cube X plus three upon five into signed to the power five AIDS minus one upon seven into SAYN to the power seven x plus c. Therefore, we get integral assigned to the power seven x d s is equals toe sine X minus signed to the Power Cube eggs less three upon five and two signed to the power five eggs minus one upon seven into SAYN to the power seven x plus the This is the final answer.

In this problem for us to find the most function. We don't even have a function perform sign. Okay, T then the antiterror to, like, Negative Warner. Okay, co sign Katie less inconstancy. So let's just buy this formula. We have seven as the constant soul. That's distracting campers here. As you can see, Jae in our case is free. So we will have a negative war over on over three times coastline data or three plus and see. And we can actually write this one as they do. 21 co sign over three blossom. Constance. Yeah.

The question would calm down. If you have an integral on the Yuba and you and by formula, we're gonna coaching uba and that's one over and just one plus e. So just in this formula here may will deal with the question. Now we have the square root of the 70 square Leske Trail DT. And here it will use the new ICO actually inside square root. And then the new ego, June fucked in T so means that DTE Coach Dean, you are over 40 now. It means then we can do the substitution now. So that's Please stay the same. This one here they were recorded a squared under New and DT becomes the dean. You ever fought in the noticed? Then we can consider it a win that day Number 14 is constant camping outside, So we have the 1/14 square into girl, no square root. We can run this Tangela Power half and then by a blind of formula Monday we add one to the power so we'll be 300 to give any between and a Jew plus c. Then we get echo to It was even if I would have won over 21 but new eco Jew, This one. So I have the 70 square breast around our seven. A trend of job the sea.

All right, So this is problem. 8.3 point 16. And we have were given this indefinitely enter or seven times her sign to seven powerful team. Now, seventh power is this Is this a little big number? So if you try to convert her sign of seven Chris, I knw 2% party into combination off bigger angles of co science going Toby Really long computation. So the trick is, as usual, we have We want to do some your substitution here. So again, the candidates or you is equal to other sign or co sign. But if you feel do co sign off t, which is the name one, then we will have to deal with, Um we have to deal with sign of tea here, which we do not have, That's why. So this means that this is not a good substitution. I mean, this will say work, but it's going to be a lot more work. So what we do here is we do the other substitution. So the good substation is U equals sign. So then do you will become course I in tea did Teo and we have a seven power of CO sign on going tohave one there. So what's left? This? We have Chris sign to a six power he But this isn't too bad because this is Carson squirt ti to the third power. And if sign of tears equal to you, that question and squares tears equal to one Linus, um, sign squared. He too cute. It is just one minus. You squared to reserve power. So this means we can write our internal Interval s seven times one Linus, you square, Hugh. And do you now? This is a simple polynomial of six theories. So you can. I mean, we can do this simply by expending its going to be a bit tedious, but what it's not is probably the best way we can understand Interval. So basically, if you expend this, if you remember the cubic formula for expansion, it's one minus three years squared, plus three year to fourth my issue there. Six power, Do you? And then we into it off them were individually So this is seven times you minus you, cube. And then they described the three over five. You 2 50 minus You two are seventh divided by seven plus c. So Now you all have to do is just a substitute back. No sign of vehicles you into this expression. So a student here sort of final answer here is going to be Well, I have Sometimes you just sometimes safty seven times signe off t Yeah, the minus. I have seven times, you cube, which is sometimes sign. Ah, Hugh R t. And then I have 21 over five times into the fifth, which is 25 we want do I bet five sign after 52. Finally. I have, um So there's seven castles, which is nice. Miners sign two serve tea, plus a constant. So the important takeaway from this problem is that this if you make the right substitution, then doesn't even matter if you don't know. Um she don't know. What's the formula for? What's the simplifying formula for this high powers? Off course. Einer, sign this your simply by using the right substitution, you simply turned the problem into a integration of polynomial. And then which you Khun D'oh! Very simply wait. Expanding the whole expression


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