evaluate-the-given-definite-integrals-int_-20sqrt2-x4-sqrt33-x8-d-x-3

Question

Evaluate the given definite integrals.
$$\int_{-2}^{0}(\sqrt{2 x+4}-\sqrt[3]{3 x+8}) d x$$

Dwijendra Rao · Accepted Answer

Hello friends. We have to value the given different in deal that is ingenious enough -2-0 Square root of two. x plus four minus of Saratov three explicit over the X. Yeah. Okay we can solve it 20 years. None of this will be two works plus 4 to the power of three were to a home three words multiplied with coefficient of x minus off three explosive to the power of 413-1413 multiple with Coefficient of X. That is three LTD manage to do zero. Mhm. So this will be mhm When we put the limit when I actually close to zero this will be For to Devour three Y 2 upon three minus Here to the Power four Y 3 of 14 minus off. When we put except cost to manage to Then first time will be zero minus off six plus it To the power of four Y 3. Okay of one thought. Okay so this will be a cost to eight upon 3 -3 Here to develop 4 1314. So that is because too It to depart for 1 3. So this will be to the bar for all of that is 1614 minus of that is bliss. So this will be To the power of four x 3 upon four. Heard by 3- or four plus. To the power of four That is 16 to the power of one x 3 upon four. No we will calculate it with the help of calculator. So eight upon 3 minus four plus 16 to the power of one x 3 upon four. So this will be a costume Yeah minus off 0.70 33 So this is the answer I hope you initially thank you.

Amy Jiang · Answer

we felt that we career eight as to square root of choose word. I know, um, it&#x27;s that X to be equal to choose for route to tangents of their. And now, if you take the X, that&#x27;s equal to two square root, too. Chickens squared. Rosetta data extenuating. This three yet to square root of 23 part three. Thanks. Looks just leave it like that. And then we have changing to the power of three data, and then we have the square root. We can actually factor out, um, to square root of two. So we have a choose group too, And then we have discovered with of changes squared plus one so that&#x27;s equal to seek it squared data. And then I also have DX, which is to square root of two. You can square the better you get. Okay, so that gives me the integral of two square root of three fr of 345 And then we have challenging to the power of three Dido and then, um, chickens to the power of three data data. And I would see, um, why don&#x27;t we Let&#x27;s you be equal to seek it of the drugs we know that the U is equal to seek it of data tensions of good um, be data. And I could just rewrite my remaining 100 squared as sequence. That is, we have tended growth, which 100 squared is equal. Teoh, she can squared minus one. And then I have, um he can square to beta and then I want are meaning seeking to data Can engines of beta they dont judicial abortion Here is to you analogy pace are seeking with you were there in a role of to three per square but a three to power five And then we have you squared minus one use word and then you so it&#x27;s pulled out that constant of Ari&#x27;s. And then we have you to our poor minister to park to do you. So that&#x27;s equal to two square root of the para five. You should par 5/5, but a par 3/3 and I&#x27;m replacing are used you if you could seeking a bit um so we get to square 3 to 5. He can do par five Bill over three. Sorry. It&#x27;s fine. Minus He came through power to Ugo over three and then we have crusty. And I don&#x27;t recall that you can&#x27;t will. Um let&#x27;s draw a triangle. You know that X over. Choose grove it of to the detentions. Do we have ex cuse Gertrude of two? And then, um, this is the square root of X squared plus eight. So seeking is the square root of two for ex were close eight. It&#x27;s no bad seeking data in terms of X square root of X squared plus eats over the square root of two times two. Okay, so we get this as our following solution. You just need to replace this term with this time.

Kelly Brooks · Answer

and this integral. We have a sum of squares that is included that would imply to use triggered a metric substitution. So start by trying a right triangle in certain ankle fada. And since we have a X Square plus eight, that implies at the square root of X Square, plus eight has to be behi part news because that would be a sum of squares, implying that the sides would be the square root of each of those terms. It was word of X squared the axe in the square root of eight, the other side, which we could also write as to score little to. But for now, we&#x27;ll leave it a square root of eight and will simplify leader. So now we can use a strangle to build our substitution. We want to use a constant side and the excite so that we can buying and an expression for X. So we recognize that tangent Beta is opposite over Jason or, in this case, X over square mood of eight, therefore, making X equal to the square root of eight times tangent beta by multiplying both sides of the equation by square really next week and find the X by different shade in both sides. The constant stays the derivative attention thing that a secret squared data be data. And we can also find an expression for the square root of X Square plus eight. Using the constant side and the high pot new site, which has the square root of X squared plus eight in it and the relationship toe data on those sites would be either co sign or seek it. In this case, let&#x27;s go ahead and you seek it and we will see that Signet Data will be the high pot news invited by the adjacent side Or, in other words, the square root of X Square plus eight. But the the square root of eight times seeking data, we multiply both sides of our equation by the square root of eight. So we&#x27;re gonna use the substitution is to rewrite our integral and do recognize that we could have gotten direct substitution and for X in are integral to come up with the same expression for the sort of X square plus eight. So in this case, let&#x27;s go ahead and rewrite are integral in terms of data, but we&#x27;re going to drop the limits of integration. We could find the equivalent expressions for fate up for these given values of 01 But instead let&#x27;s go ahead and drop them for now into a direct substitution. And we know that X is equivalent to. This world would have eight times a tangent of data, and we know that the square square root of the quantity X Square plus eight is equivalent to a square root of eight times seeking data. And we also know an expression for DX to be the square root of eight trying seeking squared Payday de Dana. So now we haven&#x27;t integral completely in terms off data, but simplify the integral so square root of eight times square word eight times square root of eight. Technically, that square root of eight quantity cubed. Recalling that square root of eight is the same thing as two square root of two. That&#x27;s too cute times the square root of two square times a squarely to or, in other words, 16 square roots of two times the integral of tangent. Data tying. Seek it CUC Vada with respective data. Next, we can notice the following that if we split the seeking cube up and extract one of the sequins seeking taters off of that Seeking cute. We can find a U substitution. So it&#x27;s a rewrite. This is Tangent Beta seeking data and then seek it square data data. So, since the derivative of seeking data seeking data tangent Finger, we can now do a U substitution by letting you equal seeking data. Your next get 30 years seeking paid up tangent, beta time, state data. So now, by substitution, we have 16 square root of two times the answer pro of while the secret data is now, are you seeking data was being squared. So now we&#x27;re square that you intendant. Data seeking data D data all gets replaced with a dean you. So now we can use a power rule to integrate this integral. So the 16 square root of two was our constant. When we integrate you square with respect to you, we will obtain 1/3 you cubed plus a constant next Let&#x27;s fact substitute. So we now have 16 square root of two divided by three because 16 square with two times the 1/3 being 16 square root of two divided by three. But you now gets replaced with a secret beta. We were Cuban, the use and I were going to cube seeking data still added on our custom of integration until we replace the favors with excess and put our limits of integration back in. So let&#x27;s now recall that sixteens were rid of to over three trying seeking Cube data. Well, let&#x27;s recall that we had a substitution for secret data. So doing that substitution, we see that second data was square root of X square plus eight all over the square root of eight, and that we want to cube that expression. Since we have brought the excess back in, we could bring back in our limits of integration from 0 to 1. Let&#x27;s go back and see where those wouldn&#x27;t so integration were. See what one? The original problem, still in terms of X, is so now we&#x27;re going to substitute in the one for the excess first and then subtract off will be getting we plug in the zero. So we&#x27;re gonna leave. The sixteens were ready to delighted by three as our constant out in front one squared is 11 plus eight is nine. So I have a square root of nine. Just three divided by the square root of eight. Quantity. Cubed minus slugging and zero or substituting in 00 square plus Angus eight. So have square root of bait divided by the square root of eight, which is a one cute will have 16 square root of two, divided by three times Expression. Three. Cubed is 27. Recall again that the square root of eight cube was essentially the square root of eight square times the square root of eight or eight times two square root of two with 16 times forward to, so I have 27 divided by 16 square root of two minus one. I&#x27;m going to next distribute the 16 square root of two, divided by three to each turn just for rewrite. It will notice that the six teams were really choose well. Reduce out in the first term in 27 divided by three will give us a nine, and they will still have minus 16 square root of two divided by three. That&#x27;s one option has an answer. Another option is to find a common denominator between these two terms. My common denominator would be three. So 27 divided by three is the same thing is nine. So we could also write this as 27 minus 16 score little to all divided by three. There&#x27;s another option for an answer, and yet another option would be to divide out the 1/3 times the quantity 27 minus 16 times forward to and then yet another possibility. If we want an approximation, we can enter the center calculator to see that her approximation would be about one point 4575

Amy Jiang · Answer

So since we have a square root of this insane term, we&#x27;ll be using yourself. Well, this is of the form acceptable to four fine vato. So let&#x27;s, um, rewrite this as X over four is equal to sign a bit. And I&#x27;m looking drop our triangle in case we need it. So opposite over I Pont news, the remaining it is four square that 16 minus x squared. Okay, So what do you write this? So we have the square roots of four minus x X is four squared. Actually, that should be two squared, right? Yeah. So this election for and this is for as well, yeah. So yet four sine squared. And then Well, that&#x27;s fine. DX, that&#x27;s equal to to co sign up there. Yeah. So yet we contract out of four. The square root afford to do we get to square roots of one money signing squared of data. So that&#x27;s coastline square. DaVita my art degree in identity. And then we have DX. That&#x27;s, um, to so that&#x27;s four whole sign of data. You data. Okay, so taking the square root, I get coastline. We stopped two of them to get four co sign squared of data. You data. Okay, so I&#x27;ll use my half uncle identity so I can write this us for one plus co sign too. Data over to the data that gives me to in a girl of one plus co sign of two data Ubeda and I&#x27;m taking our interval. We get to see the times they don&#x27;t plus sign of two data over to. And now we want to about weight that are at our endpoint. So let&#x27;s actually change our endpoint. So we have when actually good to we have sought to sign up to And then what x is equal to zero we have, actually, we&#x27;re solving for theater. So when exiting to we get to sign, they don&#x27;t. So that&#x27;s one is equal to sign of data. So data is able to sign inverse of one. And then we also get better is equal to, um let&#x27;s see what x zero, we get signed interests. They&#x27;re all so we have sine inverse of one on sign, Injurious of zero. Okay, so yet, to our data is sine inverse of one plus fine of two sine inverse of one over two. And then we get minus two times fine. Inverse of zero plus sign of two sine inverse. Observe over two. Okay, so we get before execution.

Amy Jiang · Answer

Okay, so let&#x27;s stop. I&#x27;m ready for the perfect square. That&#x27;s 2 to 2. A nomination A square root. We have the form of chickens. Side is X is equal to two seasons of data. And how much find TX, That&#x27;s to sequence of era Qianjin data data analogy Replace what we have. So we have to, um de exes out two times two So that&#x27;s four begins of data Attendant of data the data and then we have over and square roots of that&#x27;s for Seacon Square to beta minus four. We contract about our four When we get the square root of force, that&#x27;s too. And then we&#x27;re left with the square root of seeking squared minus one. So that&#x27;s tended squared. And in canary, actors of dependents. Okay, so are challenging cancels on our two and one of our shoes. Cancel here and we&#x27;re left with the integral of to chicken of data. And we know that the intervals chickens that&#x27;s 54 through, um natural log of si jin&#x27;s there plus conscience. And then we have plus c. Okay, so let&#x27;s draw out our triangle. So if we have a triangle with angle ghetto X over to Z could chickens. And that&#x27;s equal Teoh one over coastline threats to us. And this is two squared where? X squared minus, boy. Okay, so we actually does two times natural log of sequins, which is I over to plus tangent, which is the square root of X squared, minus for over two and then fusty. Okay, citizens are following solution.

Vipender Yadav · Answer

here in this problem, we have evaluated the interval zero to X cube on little X Square plus one, not in the eggs. Now here I&#x27;m substituting X us to danger Tito. So you get Deeks as to seeking square Tito Digital. And when X is zero, we get Tita as zero. And when X is too, we get Tita us then inwards. One riches by wherefore so the integral becomes integration. Zero to buy by phone on Dhere to come two. Time to cube time Staying in Cuba. Tito So I&#x27;m taking toe cube outside So to become eight on under Route X Square means to square tangent square teeter and I can write four as two square times dx dx Harris to seek and score Tedo dot detailed I&#x27;m digging toe outside, then trickle so it&#x27;ll become 16. Now we can write this tome that is under root Two square tangent square T topless to square I&#x27;m taking too common so and I&#x27;m taking outside the route so we&#x27;ll get we&#x27;re left with one plus change in school Deter on one Palestinians 20 days seeking school, Tito on route off seconds Credit is sick and leader so we&#x27;ll get two dames secret. Tito, An intruder becomes zero to buy by four Danger Cube Teeter. This is two seconds later. I&#x27;m taking you outside the integral. So it is. I&#x27;m 32 on Dhere. It is sick and Tita times seeking Squared Peter Norton Meat eater. No, I&#x27;m writing this in Dragila&#x27;s started to add as it is zero to buy. Therefore ancient square Tito times seeking sweet eater and one agent TDM ready here on second theater as it is dark day, Tita. Now we can use on formula the pain it square. Tita is equal to Seacon Square Tinker minus one. So the angel becomes zero to pi by two Seek and square t down minus one times Seacon Square, Tito Times Change and Tita Times seeking Tito dot De Vito Now against substitute you as seeking Tito So you get the U. S seeking Tito Danger Tita Benedito. And when he dies by 22 0 here it should be by before not by by two. It should be by before it headed Children by before. So when Tito zero we get the U. S. One And when Tita is bye bye full! We get a U S route to? No, the integral becomes integration. We left 30 to her should target. So integration 32 is taking out. I&#x27;m doing it outside one to root two hair. It&#x27;ll come you square minus one times you square on This is Do you? Now we get a very decent trickle Simply so tardy too. Onda, let me right more clearly. Integration. Want to root two headed Come your of our four minus you square dot Do you now The sting will be calls to started to the integration off the year The power four is U to the power Fife minus u to the power three divided by three on the limit is from one Joe root toe. Not simply we have to plug the limits so we&#x27;ll get will be called to 32. First of all other playing roach to so route due to the power Fife and divide by Fife minus route to cube Divide by three. Now we have to plug a lower limit. So one by five minus one by three on this will be called to 64. Divide by 15 under route to plus one with brackets. So finally we can conclude that the Androgel zero toe X Cube Underwood X Square plus four dot dx is equals to 64. Divide by 15 brackets under Andrew route to bless for one. So this is the answer for the given into the

Problem

Solve the given problems. Show that $\int_{0}^{1}…

Problem 34 Easy Difficulty

Evaluate the given definite integrals.
$$\int_{-2}^{0}(\sqrt{2 x+4}-\sqrt[3]{3 x+8}) d x$$

Answer

Related Courses

Basic Technical Mathematics with Calculus

Related Topics

Discussion

Top Calculus 1 / AB Educators

Grace H.

Heather Z.

Kristen K.

Samuel H.

Calculus 1 / AB Courses

Precalculus Review - Intro

Functions on the Real Line - Overview

Recommended Videos

Watch More Solved Questions in Chapter 25

Video Transcript

Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus

Grace H.

Heather Z.

Kristen K.

Samuel H.

Precalculus Review - Intro

Functions on the Real Line - Overview

What do you need help with?

Textbooks

Ask a question

Courses

AI Tutor

Basic Technical Mathematics with Calculus

Grace H.

Heather Z.

Kristen K.

Samuel H.

Precalculus Review - Intro

Functions on the Real Line - Overview

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

Grace H.

Heather Z.

Kristen K.

Samuel H.

Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus