evaluate-the-integrals-int-sin-5-x-d-x

Question

Evaluate the integrals.
$\int \sin ^{5} x d x$

Ruirui Liu · Accepted Answer

So let&#x27;s look at this sign. Thief&#x27;s. Yeah. So first on the right, please want science. Worst sign ends? Yes. And with them Spying. Spare axe sign? Yes. Well, it&#x27;s small. Admiring them closely. Grand science. You read by a close friends lives goes on a forced expires Spirits months won&#x27;t A glass wall? Yeah, course my insects. Yeah. Term as hey Course axe. Honest off. Send you called X forms. You know, these came to you? Us course, perhaps fall or you know them.

J Hardin · Answer

this problem is from Chapter seven section to problem number forty four in the book Calculus Early. Transcendental. Lt&#x27;s a condition by James Door. We have an integral of sign. Next time, see cancer the fifth power here weaken. Use the fact that she can&#x27;t is one over co sign. So let&#x27;s pull out One factor of C can right is one over co sign and then we have our remaining seeking to the fourth power left over. So here we have integral of tangent time seeking to the fourth power of X. Next we can go ahead. And since we already have one factor of tangent here, let let&#x27;s pull out another seeking from the Sikh into the fourth. So we have tangent times. He can&#x27;t times he can. Cute. This is just we do a u substitution. Let&#x27;s take you to B. C. Can&#x27;t then do you is seeking time, Stan, which is precisely in our incident we have here. We have tangent times. He can&#x27;t gx so using our new substitution, we can rewrite this as you cube coming from C Can cube. Do you? And then we could use the power rule to evaluate this. You to the fourth power over four. Plus he and finally, we should go back to our U substitution to replace you with Seek Innovex. So we obtain. She came to the fore Power Vex over four. Plus he and that&#x27;s your answer.

Amit Srivastava · Answer

we have to a degree Sign food, banks, private speeds, hope Sign. It&#x27;s one minus all spirits or five bricks. Demons? No. Was five picks minus those seven. It&#x27;s say, next. Deitz, you is a good oh sets on Differentiating you is a little minus sine X brilliance. Now I easy could do minus integrate you fight minus you seven. We get you six of bone six minus. You ate upon a Let&#x27;s see it become course well, but e ex upon see e minus six x upon seats. Let&#x27;s see, isn&#x27;t it?

Robert Daugherty · Answer

So the indefinite integral sign cubed of X over the coastline fifth of x d x. So the key years If I can see a function and is derivative in the integral, that will simplify things for me. So what I can do here is I knew that the derivative of co sign is minus sign That air of the sign is co sign. But I like the fact that the sign functions in the numerator. So what I can do here that just work with us a little bit? Let&#x27;s write this as the sine squared of x times, a sign of X over co signed to the fifth of X t X. Now remember, from our the Pythagorean identity sine squared that&#x27;s just one minus the coast sine squared over the co sign fifth of X t X. Now, if I were to let you equal the co sign of X, then d&#x27;you is equal to minus sign of x dx, so that transforms this integral. So if I if I were to see a minus sign with this sign function than they would have a negative in front of this interval. So this community they inter grow up one minus. You squared over you to the fifth, Do you? So this is I could get rid of the minus. Sign it. Just say that this is the integral You squared minus one over you to the fifth, Do you? And let&#x27;s just break this into two intervals. So this is the integral of one over you. Cubed d&#x27;you minus the integral one over you to the fifth, Did you? And so when you integrate this, you&#x27;re going to get, um, in this case, integrate. Well, let&#x27;s just write it like this. You to the minus three D u minus interval. You two of minus five, do you? So when you integrate this, you&#x27;re going to get, um, here, you&#x27;ll get you to the minus two. Divided by negative twos. You could minus 1/2. You squared, and then you have plus 1/5. Excuse me. Went over. You integrate this, you&#x27;ll go to the money, you a bit of the fourth power and then divide by negative for which is right there, plus a constant of integration. And so what we see here is remember, substitution you You was the co sign. So this gives me my answer of 1/4 co signed to the fourth of X minus 1/2 co sine squared of X plus a constant of integration. Now, let&#x27;s just take a look at this integral once more to illustrate that, um, with integration, sometimes you can get a different answer. That is equivalent. So let me just stay with the same color here. Um Original Integral was the sign cubed of X with co sign to the fifth power of X DX. Listen to rewrite this as signed cubed of X over co sign cubed of x co sine squared of x dx What we know from Trig identities which you will use a lot in integration. The sign over the coastline is just the tangent. So this is the integral of the tangent, cubed of backs and then one over The co sign is the seek. And so this is seeking square toe vax d X, and any time you can see that combination of tangent seeking squared, you&#x27;ve got something that you can work with. Um, here, if I were to let you equal the change, it of acts in de U is equal to the secret squared of X DX. So this is just the integral you cubed, do you? Which is you to the fourth over four, plus a constant of integration. And so what this turns into this just turns into Remember us? Tangent of X. This is the change it to the fourth of X over four, plus a constant of integration. So again, um, you have to sort of be confident what you&#x27;re doing, because there&#x27;s not quite obvious. But this answer and these answers, they&#x27;re not the same, but they&#x27;re equivalent. One way to check us to differentiate. You could different change either of these and get back to with some algebraic or trig manipulation your your final answer.

Linh Vu · Answer

calm down. It was under indigo on the sign you do. And then the answer It could, you know, minus course on your policy And in this question were given in the girl. Sigh under five over. Thanks. Dividing by the x square T X Now, in with just that, you go to this quantity and then do you go ju minus five over X square the X So it means that the X over X squared and we exactly ico jude uh D year over minus five. And therefore we can be able to read this integral in terms of that you. Now so have the sight of you and terms with do you over minus five. The Mantis Fiza constantly camping outside into girl. So getting co chairman is 1/5 now site you do you buy formula here. We should get the minus coarse sand and you plus C. But you got your five x So echo Judah, one of the five ghosts I under 500 X blessing

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.

Problem

Evaluate the integrals. $\int_{0}^{\pi} \sin ^{5…

Problem 7 Easy Difficulty

Evaluate the integrals.
$\int \sin ^{5} x d x$

Answer

$-\cos x+\frac{2}{3} \cos ^{3} x-\frac{1}{5} \cos ^{5} x+C$

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$-\cos x+\frac{2}{3} \cos ^{3} x-\frac{1}{5} \cos ^{5} x+C$

Thomas Calculus

Grace H.

Kayleah T.

Kristen K.

Joseph L.

Integration Review - Intro

Definite vs Indefinite

George B. Thomas Jr.Thomas Calculus

Grace H.

Kayleah T.

Kristen K.

Joseph L.

George B. Thomas Jr.

Thomas Calculus