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(a) Use the reduction formula in Example 6 to show that$$\int_{0}^{\pi / 2} \sin ^{n} x d x=\frac{n-1}{n} \int_{0}^{\pi / 2} \sin ^{n-2} x d x$$where $n \geqslant 2$ is an integer.(b) Use part (a) to evaluate $\int_{0}^{\pi / 2} \sin ^{3} x d x$ and $\int_{0}^{\pi / 2} \sin ^{5} x d x$(c) Use part (a) to show that, for odd powers of sine,$$\int_{0}^{\pi / 2} \sin ^{2 n+1} x d x=\frac{2 \cdot 4 \cdot 6 \cdot \cdots \cdot 2 n}{3 \cdot 5 \cdot 7 \cdot \cdots \cdot(2 n+1)}$$

a. $\frac{n-1}{n} \int_{0}^{\pi / 2} \sin ^{n-2} x d x$b. $\int_{0}^{\pi / 2} \sin ^{3} d x=\frac{2}{3}$ $\int_{0}^{\pi / 2} \sin ^{5} d x=\frac{8}{15}$c. $\int_{0}^{\pi / 2} \sin ^{2 n+1} x d x$

Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 1

Integration by Parts

Integration Techniques

Missouri State University

Campbell University

Baylor University

Boston College

Lectures

01:11

In mathematics, integratio…

06:55

In grammar, determiners ar…

11:18

(a) Use the reduction form…

08:52

03:34

Use the Wallis formulas in…

04:12

Rewriting Integrals

05:33

09:56

03:21

A sine reduction formula U…

03:04

Given that $n$ is a positi…

06:09

(a) By completing the squa…

15:02

(a) If $n \geqslant 2$ is …

four part, eh? We recalled a reduction formula. That is the one that we have here. So if we want to compute dainty were from Cyril, they were too. Sign to the poor orphan effects T X. We just need to evaluate the right hand side of the reduction formula, meaning this part here. So we get minus one over end course and effects, saying to a power off in minus one x and we need to evaluate between civil on power too, plus 10 minus one over end 18 to grow from cereal. A word to him, upside to the power of Finn minus two affects Deeks. So if we have always the first part, we will get the co sign off by over two times assigned to a polo fan minus one. Never would play over too, minus the same function, but now evaluated at the lower limit that this case is serious plus and minus one over, and they interrupt from Syria to pay well, too off signed to the power of friend minus two x t X and we know that course. And if I were to zero and sign off, zero is zero. That means that this whole expression here in this the whole term it's going to be serial. So basically, we get that thing to grill from zero to buy over too upside to a power. A friend of X dx. It's equal to n minus one over and vein tumor from seal to play over too. Sign to the power of Finn minus two affects you for part B. So we need to evaluate first. You seem too well thing too well from Syria to play word too. Signed to power three off X dx and we need to use the formula. We get in part a meaning this formal here, and in this case, we need to use it with n equal to three. So if we replace in the right hand side with a three minus one over three name to her from zero by over two, sign off three minus two off X GX. That's going to be to 1/3 thing to grow from cereal to play, too. No sign of X t X and us to third and the interviewer off sign is minus school. Sign off X, and we need to evaluate this in between Ciro and Pirate Joe. So we get to third times minus the coal sign off. Fly over to the upper limit. Minus minus. CO sign of Syria. Co Sign off Baylor to a serial and co sign of Seo is one and minus time. Minus is once we get just to over three. Also, we need to evaluate the integral from zero too. But you were too off signing to the floor. Five Health, ext. D X. And again. We can use the formula that we're getting park, eh? In this case with and equal to five. So it would take a look. Here. We will get five minus one over five. Thing to go from Cyril. Toe player too up. Sign off. Five minus two. Thanks, Deeks. So this will give us four over five times interior from Syria to pay over to off. Sign to the power of three off X D. X. So we need to evaluate this integral. This will give us for over five. See, you will to pay over too. And we can rewrite the sign and kill us. Sign off pecs time saying square effects DX. So we get for over five things over from Syria to play word too inducing. So, trigonometry identities. We'll replace science where? One minus close and square Fix T X. So we get four over five. I think they're from Syria to play word too. Oh, sign off. X minus. Sign off. X coursing square. Both x ttx. And this is for four or five the integral of sine. We know that it's minus clothes and effects. And if we use this substitution, you equal to cosign effects. We can solve this integral to get course and cube off X over three. And we're below eighties from zero to pirate too. So that will give us for over five. First, we will await in pine over two and we'll get minus course I by over two plus course and cure flavor too. Over three minus on this course in at zero plus course in Cuba. Zero over three. Okay, so we know that this is zero and this is also zero. And closing off suit was one only. This one. Here's one. So we get four or five times and minus 10 miners will give us one minus one over three, and that will give us four or five times toe over three. It is the same. That eight over 50. And that's part B. Okay? And now we need to finally do part see, and went to use the formula in a I'm gonna call before me today, so we have it in hand. We need to use this formula to prove the following the being together from Cyril to play word tool off sign to the power of 20 plus one affects E x A simple too. Two times, four times, six times, two in over three times, five times, servant times two plus one. So if we start with the integral and we applied for me that we're getting a ones while we will get peace 20 plus one minus one over 20 plus one time Seeing to grow from Sewell Tau pi over too. Sign off tour plus one minus two off x d. X. So wasted the every right, this formula by changing and with toe and plus one and we'll get this. Okay, So this is the same us to win over to Empress one times integral from serial tau pi. Over too. In here with that sign off. Tow n minus one off x t x and we're gonna play this formula one more time. Now, Toby seem to go here. Who would do it first? Now and what we get now, Uh, we use the same formula, but with an replaced by twin minus one. So we get 12 minus one minus one over twin minus one time. Seem to grow from Ciro to buy over too. The sign off too. And minus one minus two off X dicks. And that's just to win times to one minus two time toe in my plus one, too. And minus one in this times integral from cereal by. Over two off. Sign off to the power of twin now minus three off x d x. So we see that every time that we apply the formula to the scene to grow we have uneven power tow the nominator and the north power to eliminator and scenes original exponents off the sign is too and plus one we can apply this formula end times and if we apply it in times we will get that thing to grow from Syria to play over to off sign off 20 plus one effects T X is going to be well, too. And times two and minus two times tool and down We will have in the denominator 20 plus one toe and minus y, and we will follow it to work it to three here. And he would get into well, to pi over too, the sign off to end. So since we apply end times of formula tow the exponents here that originally was to n plus One, we need to substructure two times. And since each time that we apply the formula works of threat to that means that here the exponents should be equal to one x d x, and we already compute this in part B and we know that it's one. So we got that desire formula by using the formula that were obtaining

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