evaluate-the-integrals-int_132-x-6-5-d-x

Question

Evaluate the integrals
$$
\int_{1}^{32} x^{-6 / 5} d x
$$

Nathalie Luna · Accepted Answer

soon in this problem, we are asked to evaluate this integral. And so the first thing that we notice spotted it&#x27;s that it&#x27;s an integral off the power eggs and were derived integrating with respect to X. So we the first thing that we&#x27;re going to recall this tower pour into girl. So for integrating X lay still power and where in has to be different from one. Uh, this tentacle will be ex raced and plus one and we divide bye and plus one to this is a indefinite And the growth we are, eh? A constant in this case, we don&#x27;t really need it. But so in our case and ISS minus sixty five. So first thing that we want to do to integrate this pendant in plus one. So since this is a fraction of going to do it separately, yeah, and plus one isthe over six from seven hundred six over five plus one. But instead of range one since we want to be able to add a two two numbers were one of them is a fraction they have to have the same the number denominator. So we want something with sign on the bottom Since we&#x27;re talking about one, we want same number asked in a greater and greater. So instead of writing one working to write five five. Still, this is equal to five days right here and now top arguably. Haps my six plus fine. And that is people too. Minus one over. So this isthe class. Fine. Which is what we&#x27;re gonna do. Okay, uh, look. And here and here are integrated. So and this first up, we don&#x27;t need this practice. We&#x27;re already integrating, and we have ex place to n plus one, which we&#x27;ve figured out this minus one over pi Delighted by the scene number Mendes one over. And this expression, it&#x27;s going to be when x equal one and up to thirty two. So before, without practice, huh? Function in thes and points of the integral. I&#x27;m home to reeling to this so that it&#x27;s easier to work with. So we have, uh, this minus saying we&#x27;ll keep it here. We could put it instead over here instead of here. And this five since it&#x27;s in the numerator after the narrator, we can bring it back here to bring it up floor here. All right, so we can rewrite it like minus five eggs. Fruit base, too. Mice one. Over and again. We don&#x27;t lose this X equal to one, two, thirty two. All right, so now, before we do this next part, which issue seemed the fundamental Purim of calculus where we evaluate this when ex house and one x thirty two minus the same expression on except one we are going to I remember some of that fact. So here we have. When we raise one to any power, get one and thirty two, we can write that us to the park soon. I am going to first one X equals to this first part is when X equals thirty two, which we&#x27;re going to raid us two to the fifth power and then we&#x27;re going to use the same expression when X equals one. All right, so let&#x27;s move this downtown. So we have Manus fire. That&#x27;s a constant Tim Specs, just two to the five. And this text is going to be raised to one. Men&#x27;s went over and bring to this is the expression on X equals thirty two and three Subtract the same expression on X equals y. So the constant does not change eggs. This one and the power. All right, so you know, it&#x27;s all constants. We can no work on getting this a little bit. One similar simplifying this right? So we have. And this power, it&#x27;s five times who won. Minus whatever fight the chain. Since this is denominator, the numerator and denominator thes cancel out. So we&#x27;re left with two races to the minus one, which is people to one half. And in this case, since we have one race to some power, this is going to be equal to just so this simplifies too. Minus five times one half between kin, Right? Like this. And this place no, outside this minus and minus turned into positive. And this is five times one. The chains. It&#x27;s just five. Now again, tow up to numbers we have. This is a fraction and a hostin a mayor too. So we&#x27;re going to put it to here. And since we do it on the bottom of this number, we&#x27;re goingto have to do it the same. No matter. So think of this five as five over one. That&#x27;s what it is. And then we&#x27;re just multiplying by to both the numerator generator. Right? So now we can actually at these numbers till you have minus O on the bottom. We have to. That does not change. And we have minutes five plus times two. And that&#x27;s equal to five. Number two seven. That isthe our answer when we evaluate this tibia.

Patrick Vaughn · Answer

in this problem, we have the integral of one over Exodus six times X to the fifth plus four DX. So the point of this problem is we want to manipulate this in a way that we can start using partial fractions. The reason why we can&#x27;t really use partial fractions now is because of this extra the fifth plus four term. We don&#x27;t really have a good way to factor that. So we&#x27;re gonna want to play with this a little bit. So suppose we let u equal Teoh extra 50 plus for Because it&#x27;s kind of the problem child here. So then d you would equal five extra before uh, D X. Okay, So what would happen if we multiply a top and bottom of our fraction by excellent forth. So that would get us exit the fourth over x to the 10th times X to the fifth plus four DX. So this is kind of what we&#x27;re looking for. So notice that what is X to the fifth equal to so X to the fifth is equal to U minus four. So here we have an extra fifth squared. Um, here we have a you and here we if we put the d X in the top. So here we have a 1 50 You kind of get us what we want so we can rewrite this head roll as 1/5 d&#x27;you over u minus four quantity squared times You all right? So now we&#x27;ve got the problem into a form where the Inter Grant is a polynomial divided by call. No meal so we can use partial fractions. So we&#x27;ll have one. Let&#x27;s not use the 1/5 Let&#x27;s pull the 1/5 outfront just cause it&#x27;s easier doing math with one seemed easier than doing math with. So have you minus four squared you is gonna be equal to, so we&#x27;ll have some constant over U plus another constant over U minus four plus 1/3 constant over U minus for quantity squared so we can use the heavy side method to find A and C. So then a is gonna equal one over negative for square. So it&#x27;s gonna be 1 16 c is going to be one over for So Okay, so now we just need to find be on. We&#x27;ll be there. Um, so let&#x27;s go ahead. Multiply the left and right hand sides by the dominator of the left hand side. So we&#x27;re gonna get one is equal Teoh A No, we don&#x27;t have any day 1/16 times U minus 41 squared, Uh, plus B times you times u minus four uh, plus CC was 1/4 you. So far so good. Okay, so we need to start pulling things that Arby&#x27;s and everything else out. Um, if we just look at that, Constance will have an easier Actually, we just want to look at the U terms or the U squared. I lets go for you squared. So we&#x27;re gonna have being used Squared here will have 1/16 you swear is equal to your zero use words. That means that B is equal to negative 1/16. So we got all the pieces. Let&#x27;s rewrite our generals are Interval is going to be. We have a 1/5 in front. Don&#x27;t forget that. What was a a is 1/16. So we&#x27;re gonna have ah 1/16 1 of the you. What was BB is negative. 1/16 1 over U minus four. And finally see, So we&#x27;re gonna have plus 1/4 1 over u minus for quantity squared, do you? Right. So from here, we need Teoh for this term we&#x27;re gonna have to do a substitution is we&#x27;re gonna be equal to U minus four. And for these two, we could just integrate them directly. So have won the times 1/16 l an absolute value U minus 1/16 Ellen Absolute value U minus four. Uh, but some constant. Um, plus 1/20 5th Groll of one over the square. Davey. So far, so good. Okay, so we have one. We can combine these. So 1/80. Who? That&#x27;s five times 16. Ellen. Absolute value of you over absolute value. U my passport. There&#x27;s a constant completely integrate. This will have minus 1/20. Uh, one over already. Got the constant. Okay, So what was the V was U minus four. So 1/80. Ln absolute value U over U minus four, um, minus 1/21. Over U minus four. Awesome concept. What was you? Let&#x27;s go up here. You was equal to X the fifth house for So let&#x27;s go ahead and plug it in. So I have one over a l. An absolute value extra fifth plus four close absolute value all over. So if we have excellent fifth plus four minds for that&#x27;s the same as extra The fifth, uh, minus 1/21 over X the fifth. Sloppy. Thanks. Here we go. Brought some constant, and that completes their come.

Tom Greenwood · Answer

okay, Evaluating this integral in The first thing we have to do is take the five X over the X squared minus X minus six, and you are partial fraction decomposition of this. So what we need to do first ISS to factor the denominator and realize that that factors into an X minus three times X plus two and those are the denominators off our decomposition. And then those being linear and not repeated, we would have just a over X minus three plus B over the X plus two. Now we multiply all that by our common denominator to get rid of fractions. So we just get five x on the left hand side. Each of these denominators will reduce, so we&#x27;re left with the numerator times. The other denominator. So eight times the X plus two se x plus two a and B times the X minus three his B X minus three b And now we can equate our terms on inside quickening coefficients. The five x five gotta equal the A plus B x so five equals a plus. B is one equation it would have and then are constants to a minus three b well knows, add up to zero because there&#x27;s no constant on the other side. So now pick whatever method you want to do to solve this system of equations. Whichever method you&#x27;re using, you should end up with a equals three and B equals two, and that should be a minus three b non A plus three b typographical air there cause it was a minus three B. All right, so there&#x27;s our correct system and three into early. It&#x27;s the correct solution here for a partial fraction decomposition. So we have the integral from negative wanted to of three over X minus three. Pause to over exports to the X so we can separate thes and individual and girls. So we&#x27;re gonna have the girl from negative wanted to pulls a three out in front only that one over X minus three D X and then plus again pull the two out in front of the integral. We have two times an agro from negative 1 to 2 off one over X plus two d X. Because now these air teach in the form one over you. Do you? So we have three times the natural algorithm of the absolute value of X minus three plus two times in natural algorithm of the absolute value of exports to and that&#x27;s gonna get evaluated from negative 12 So now we just have to plug those values in. So we have plugging in the to three times. The natural log Tu minus three is negative. One absolute value of negative one is one plus two times a natural log of two plus two is four minus. Now we put in the negative one three times a natural of negative one minus three is negative for absolute value of negative force for Waas to times the natural log of negative one plus two is one. Well, the natural longer than one is zero. So that zero and that zero. So I have to natural before minus three times a natural before, which gives me a negative one natural log off for There&#x27;s

Audrey Fong · Answer

we want to integrate acceptable by over one plus extra past six. I have in mind this form off cry mix over ethics. The X. If I can fit that into this one, I can use the result. Long, long epics. Let&#x27;s see. So let&#x27;s try. Let&#x27;s set the denominator to be our FX Miss differentiate it. How get six extra above by So I&#x27;m really missing. Here is a six year So this is what I can do. Ah, Buddha one was six. Year was six x five years over one plus x six PXE. So my new Morita is at Primex. My denominator here will be exc so I can use the results. The result we wanted was six long one plus X equals six. Let&#x27;s see now, since one plus extra causes is always possibly I can remove my mom. So my final answer will be 1/6 long Break it one plus x Abbas six Prickett plus C

Amy Jiang · Answer

Okay, so let&#x27;s start producing that. We have an extra to fire six in our denomination and in our numerator, we have a next to power five. So we can Let&#x27;s are you be able to accept our six and then are do you that&#x27;s equal to six. Exit about five DEA announced We divide by six on both sides. We get what we have in our numerator x five d x That&#x27;s equal Teoh integral of do you over six and then we have one over one plus you. But actually, um, we should actually set you to be equal to active her sticks and plus one. So then, um, when we take the derivative, we get the same thing. And this whole portion now is you. Okay, so doing that, we get 1/6. The integral of one over you is Ellen of the absolute value of X acquire six and plus one. And then we get plus c. Okay, so we get the following as our solution

Kurt Kleinberg · Answer

he has here with another definite in a role. We have the interval from 1 to 32 of the quantity X to the negative six myths. So if you recall, we&#x27;re gonna make this go up by one. So negative 6/5 plus one is gonna be negative 1/5 because you&#x27;re basically heading 5/5. And so with also defy by negative 1/5. So we&#x27;re going to get negative five x to the negative 1/5, evaluated from 1 to 32. See negative. Five times 32 to the negative. 1/5 minus quantity. Negative five times one to the negative. 1/5. Well, 32. The fifth root of 32 is too. But it&#x27;s the negative 1/5 route, so that, too, is gonna actually go in the body. So we&#x27;re gonna end up with negative five over to, and then we&#x27;re gonna minus a minus five because no matter what you raise 12 it becomes one. So essentially, what you have here is negative. 5/2 plus now five is the same thing as 10 over to to end it with a grand total of positive five or two

Problem

Evaluate the integrals $$ \int_{0}^{\pi / 3} 2 …

Problem 8 Medium Difficulty

Evaluate the integrals
$$
\int_{1}^{32} x^{-6 / 5} d x
$$

Answer

$\frac{5}{2}$

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$\frac{5}{2}$

Thomas Calculus

Catherine R.

Heather Z.

Kayleah T.

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Precalculus Review - Intro

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George B. Thomas Jr.Thomas Calculus

Catherine R.

Heather Z.

Kayleah T.

Kristen K.

George B. Thomas Jr.

Thomas Calculus