integrate-each-of-the-functions-int-cos-5-x-sin-x-d-x-3

Question

Integrate each of the functions.
$$\int \cos ^{5} x(-\sin x d x)$$

Tyler Moulton · Accepted Answer

we want to integrate the following. The integral of the coastline. Fifth of X. Times negative Sine X. Ds. This question is testing our knowledge of methods of integration as such. We need to have a strong foundation integral calculus to solve. And particularly the question is testing our knowledge of the power rule for integration. The power rule for integration states at the integral of U. N. D. U. Is equal to U. To the N plus one divided by N plus one. That&#x27;s the constant of integration. See So if we can identify you and and do you in this problem we can easily solve the the power rule. Since the negative sign is that of coastline, it&#x27;s easy to see that U. As cosine X. Do you as negative sine X. D. X. And N. Is equal to five the exponent of our coastline. Thus we can use the power will to solve plugging in U. D. U. N. N. So integral U. N. D. U. Is equal to 1/6 times the coastline of exploration. Six power plus the constant of integration seat

Alex Johnson · Answer

in this problem, we&#x27;re gonna find the anti derivative of negative five x sign of X. Our first step is to use immigration by parts will assign you to be negative five x and d&#x27;you to be negative. Five d x d v will be sign of X DX and the will be negative co sign X and our integration diapers formula is the integral of U V D, which is equal to you b minus the integral of the EU. And that&#x27;s where we&#x27;re getting our you and DV values from okay. The integral then will be equal to negative five x times Negative co sign X that&#x27;s R U and V values. And then we&#x27;re gonna subtract the integral of V, which is negative. Co signed X multiplied by negative five x d x Her negative five. Excuse me, D x. Okay, let&#x27;s simplify this downs. We have five ex co signed X minus the integral of five co sign X d x. It&#x27;s our final answer is five ex co site X minus five sine X plus C

Amy Jiang · Answer

Okay, So noticed that we have a odd power here. So let&#x27;s start by putting this up into and even powers to sign for X I X. I think so. Recall that sine squared about close close eye in squared is equal to one of by our jaggery and I don&#x27;t need so solving first. Fine. We get one minus co science quarterback who can write this us in a girl of one minus coastline in the squared of X. And then we still have a reigning power to fine of x. The X without using Yusof with lets you people finally and into new is it? It&#x27;s a co sign. Actually, we don&#x27;t want you to be one of us. We want you to be co sign this term here. And do you get people to negative signs? There is no sign of that. Yeah. Okay, so we have the integral of 11 used word clothes and then negative bu It&#x27;s mogul quite out that power we have, we have 11 You squared times. One minute you squared one minus to you squared was used for And that&#x27;s when we get negative interest off one minus to use squared. Plus you for four. You That gives us negative. You what? To over three years to the party. And then Wyness, you did a par 5/5 and in policy, and I was Now, let&#x27;s replace. Are you with co signer Becks? Okay, so we get the fallout.

Amy Jiang · Answer

Okay, so we see that close and have the power five, and that&#x27;s odd. So to write that as a positive power. So we&#x27;ll factor out one of our coastline of exes who have close and four coats and of X and in TX. Now, let&#x27;s see, um, we want this portion to be, Do you? That means we want you to be signed. So we need to rewrite this in terms of sign. So we could do that by using are inaccurate identity, that is, we have one minus. Sorry. Weird of X. So this is still about four. So this is now about to cause this portion is coastline square back. So now let&#x27;s let you be able to sign of X and into you. That&#x27;s equal to co sign of X, the X and this is you. Okay, so now let&#x27;s rewrite dysentery of you. So we have the inner girl sign issue squared one minus. You squared to the power of to And then do you okay, so it&#x27;s multiply out. This one might misuse word. So get one minus two squared and then plus you to power for Okay, Now let&#x27;s multiply out tissue squared So we get you squared and what it&#x27;s to you too powerful. Most U to the power of six. Do you and I&#x27;ll taking are integral. We have use about 3/3 Glennis to over five years in power. Five plus you deprive 7/7. And in policy analysis, replace Are you are you is sign of X. Okay, so he gets signs to part three X over three minus 2/5. So I into power five x and then plus sign to part of servant X over seven and m plus e. Okay, so this is a secretion.

Amy Jiang · Answer

Okay, so there. So we have a pair of five here. So here at this as even Paris. So we&#x27;ll take out one of our signs. So we have signed to four and then co sign of negative two X And in our remaining sign of ICTs Yeah. So I want this portion here to be or do you So that means that you would be co sign So let&#x27;s rewrite this in terms of co signed by using our path Agree? And I didn t That is, you know, not sine squared X That&#x27;s one liners co sign squared X So we have a meeting part. It&#x27;s part of two and then we have coastline of 82. Sign of the X. Okay, so letting you&#x27;ve equal to co sign of X we have to you that&#x27;s equal to negative sign of x DX. Okay, so what knowledge replace this, um, in terms of, uh, you. So we have one minus you squared, squared and then you depart native to and then sign of X the X that&#x27;s equal to negative. Do you? Okay, now let&#x27;s multiply in. Are you two are actually supposed to put out this part here. So you get negative in a girl of one. I see you have one minus two squared cost. You two are for Okay. Now let&#x27;s wants a part out. Let&#x27;s multiply in this negative on this YouTube negative too. Should we get the integral of negative? You did in there, too. Plus to you depart to minus two. So that&#x27;s just one. And then minus did our four minus do sit ups. You squared and then we have to You okay? I&#x27;m not taking our general. We get negative U to the power of negative. Too close was that one over? Eight of one plus two you and then minus few. Cubed over three and then plus C. Okay, so that&#x27;s one over you. You is co star Innovex close to co sign of that minus co sign. Huge Vex over three. And in Posse. No, because that&#x27;s one over. Co sign that sequence to sequins till he gets sequence of X plus two co side of X minus co sign cubed the backs over three and then plus C Yes. So we get the following

Ramzi Deek · Answer

So we have X times. Go sign five x bx So we&#x27;re gonna do this by integrating by parts So I will take excess. You and DVS co signed five x So you is X Ah Devi is co signed five x So do you is one v will be ah signed five x over five So that&#x27;s signed five x over five. So let&#x27;s start by doing the following So you Devi two girl a few d v is equal to you ve minus integral of e do you? So we already have the left side So you his ex times signed five x over five minus integral of V which is signed five x over five times. Do you, which is one those will just be signed five x over five so x Times signed five x over five minus 1/5 integral off sign five x So uh, signed five x is minus co signed five x over five So this will be X signed five x over five minus 1/5 times minus co signed five X divided by five So, uh, expanding this we&#x27;ll get X sign five x over five plus co sign five x over 25. So this is our final and

Problem

Integrate each of the functions. $$\int 0.4 \sqrt…

Problem 6 Easy Difficulty

Integrate each of the functions.
$$\int \cos ^{5} x(-\sin x d x)$$

Answer

$\frac{\cos ^{6} x}{6}+C$

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Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus

Anna Marie V.

Heather Z.

Kristen K.

Samuel H.

Precalculus Review - Intro

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$\frac{\cos ^{6} x}{6}+C$

Basic Technical Mathematics with Calculus

Anna Marie V.

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

Anna Marie V.

Heather Z.

Kristen K.

Samuel H.

Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus