evaluate-the-given-definite-integrals-int_-51-6-sqrt6-2-x-d-x-3

Question

Evaluate the given definite integrals.
$$\int_{-5}^{1} 6 \sqrt{6-2 x} d x$$

Dwijendra Rao · Accepted Answer

Hello Friends. We have to find the definite in legal. That is modified 21 six and two. You squirreled off six months 2 x. dx. Okay so now we will calculate it -5-1 Square root of six months 2 weeks dx. So this will be close to six interrogation of this will be six months to X. to the power of one way to plus one upon one way to plus one. Okay. Into you know into we went up one managed to Okay And the limit is -5 grand one so we can solve it six months 2 weeks to the power of three were to upon Mhm 32 Dumanis too -5: one. So this will be close to six six months 2 weeks to the bar. Aww three where they were born, monastery. Okay so now we will put the limit So this will be minus of two six months to the power off 3 2 months off six minus two in 25. That is 10 To the power of three x 2. So this will request two months of 26 months to that is full for to the bubble three x two months of 16 to the power of three x 2. Minister. 4 to the power three vital will be yet 16 2. Power three word will be 64 64 -8 Will be 56 Into two. That will be costume 112 So this is the and that I hope you understood. Thank you.

Ernest Castorena · Answer

we&#x27;re going to use partial fractions. Will have a tendency Times X plus two. Yeah, it looks good. So there&#x27;s a building near factors. We&#x27;re gonna have a on Rex minus C plus B over X plus two on the equations that resides to law. Here&#x27;s the situation. Any times exposed to be times X man thing end up by eight plus B should equal high and also two and minus A B equals zero. So we cannot supply the situation by three and add it to the other to eliminate the bottom. So we end up with that&#x27;s a 15 on the right hand side to be cancelled. Yeah, five it. So we have that a is equal, Teoh. Fine. But if is equal physical ability What a crazy close. The three inlets. Like an equation being my sweetie bulls to. So we have that. This animal is the same, uh, taking the interval of dream over X ministry. Plus, what&#x27;s it to over at lusted the X. We take this, we&#x27;re gonna get the times the natural log, Lex minus three plus two times and that you&#x27;re alive. Uh, exposed to now. We could We wanted Teoh bring them together. They have bringing the coalitions and its power, But this looks good

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.

Amit Srivastava · Answer

I could do integration off DX upon 36 effect a common here it will be a six Xs guy minus Piper six. We have taken six has outside but by six. So final part will be under root X squared minus pipe upon six bullets As we know in Dickerson off and I dont x squared minus a square will be love X plus on Darrelle their square minuses. So one by six look express and adult That&#x27;s the squared minus five past six for the square.

Alex Mangiapane · Answer

So let&#x27;s evaluate this ingrown hair the ex with partial fractions because it&#x27;s noticeable that the denominator of this integral is act arable into X minus two times X minus three. Which is why I&#x27;ve laid it out like this. So, um, we can we can rewrite this as a X minus three a plus B X minus two minus to be, uh, so that&#x27;s the numerator is gonna equal to X minus one. And so we notice here that the coefficients of the ex terms have to be too. So that&#x27;s right, that a plus B is gonna equal too. And also these constants three a plus two bay. We&#x27;re gonna have to equal this equals one. And I got rid of all the negative science, but you could just as well write them in. And so we can We can solve this. It&#x27;s pretty simple system of equations. I&#x27;ll do it. Ah, I&#x27;ll do it by just substitution. So a close to minus B so six minus three b plus two b A equals one. So negative b equals negative five. So if b equals five, then a must be negative three. So that works with our first equation in just a check. It works with our second equation as well. Negative nine plus 10 is one. So that makes it pretty simple to Silver. Integral. We just plug in negative three here and five here. And we&#x27;re doing this integral the ex. And so our solution just becomes five ln X minus three minus three Ellen X minus two plus C. There we go.

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

Problem

Evaluate the given definite integrals. $$\int_{0}…

Problem 26 Easy Difficulty

Evaluate the given definite integrals.
$$\int_{-5}^{1} 6 \sqrt{6-2 x} d x$$

Answer

112

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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