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2. Evaluate V9 - 43"dr_For f(c) set up An integral t0 compute the [0,1j: 'Then evaluate the arC length of y = f(r) for € in the interval integral,Ev...

Question

2. Evaluate V9 - 43"dr_For f(c) set up An integral t0 compute the [0,1j: 'Then evaluate the arC length of y = f(r) for € in the interval integral,Evaluate (tan(2x) + tan" (2r)) dr:5. The secant reduction formula says that for n # 1, sec" (c) dr sec" -2 tan(x) n _ 2 nl _ sec"-?(r) dz: Use integration by parts, with correct: sec"-?(c) and dv sec? (z) dz to show that this formula

2. Evaluate V9 - 43"dr_ For f(c) set up An integral t0 compute the [0,1j: 'Then evaluate the arC length of y = f(r) for € in the interval integral, Evaluate (tan(2x) + tan" (2r)) dr: 5. The secant reduction formula says that for n # 1, sec" (c) dr sec" -2 tan(x) n _ 2 nl _ sec"-?(r) dz: Use integration by parts, with correct: sec"-?(c) and dv sec? (z) dz to show that this formula



Answers

Use the Substitution Formula in Theorem 7 to evaluate the integrals in Exercises $1-24 .$
a. $\int_{-\pi / 2}^{0}\left(2+\tan \frac{t}{2}\right) \sec ^{2} \frac{t}{2} d t$ b. $\int_{-\pi / 2}^{\pi / 2}\left(2+\tan \frac{t}{2}\right) \sec ^{2} \frac{t}{2} d t$

We have the integral from negative pi over to two pi over two of square root, one minus sine squared, Mhm times. Cosine, DX. Now we're going to let Why equal sign of X Now it's important to note that Oh, that this substitution is 1 to 1 on the interval from negative pi over to two pi over two kind. We can also calculate d y. That's going to be cosine, x times, DX and now we can rewrite this integral in terms of why so when sign sign of negative pi over two is equal to negative. One sign of pi over two is one and then we pull again. One minus y squared, Do you? Why in this? Integral corresponds with a picture of a circle with radius one but the top half. So it's equal to one half pi r squared where r is equal to one just part about it by two

Okay to integrate this function. What I noticed first is that I have a fated to the three halves. And if they did to the one half, and I know that the derivative of data to the three halves hesitated to the one half in it. So that tells me I need to make you be the stated to the three halves. And then the square root of that is gonna be part of D U. So let u equal data to the three halves, then do you? Is three house data to the one half d theta. So we have the data to the one half right here. I don't have a three have. So I'll just put one. Went out in front here, backwards, the inverse. All right, Uh, if they d equals zero than you equals zero to the three house, which is zero, if they do equals the cube root of pi squared. That's pi to the two thirds than you equals pi to the two thirds to the three halves, which is just pie. All right, so I'm integrating two thirds zero pie. Cosine squared you and then this and this That's do you okay? do integrate the coastline square to have to put in an identity. So let's do that. Now. The identity is one plus the casino of to you over to do you. So this too and this to cancel out. So now I have one third the integral jeered a pie one plus the cosine of to you, Do you? It's a one third zero pie one, do you? Plus one third zero pie co signed to you. Do you can. I did that separately so that you can see what happens with this too. Right here. But here we get one third you from zero pie. Plus, here we get one third. I'm gonna have to make another substitution. I'm gonna let w b to you. And so d. W is to do you. So I needed to in there, which will put a one half out here. Okay. If U equals zero, then w equals zero. If you equals pi, then w equals two pi. So we have 0 to 2 pi co signed w d w. All right, so here we get one third pie. Here we get plus 1/6. The inner girl. The cosine is the sign up w from 0 to 2 pi and when you plug both those in, you get zero. So one third pie should be the answer to this one or pi over three.

Okay on this problem. They told us to let X equal to sine theta. That means in place of DX we're gonna put to co sign data deed data. So let's make this institution here in place of DX to cosine theta d theta over in place of X squared, we're gonna have four sine squared data and then in here we're gonna have four minus four science core data. Okay, so let me work on this over here. Four minus four sine squared data. I'm gonna factor the four to the front one. Minus sine squared data is the co sign of data. So we got four coast is the cosine squared of data. So over the square root of four cosine squared, which is to co sign data. So now we have to co sign data D data on the top four sine squared times to cosine theta on the bottom. Those two coastline status canceled. So now we have 1/4 the inner girl one over the sine squared d theta Kim. But one over the sine squared is co Seacon square. So this is 1/4 the integral co Seacon Square Data Dita So co ck in squared is theano tie derivative of the co tangent. It's OK, minus the co tangent. So you get minus 1/4 co tangent of data plus C. All right, now we have toe switch back. We can't leave the answer with data. We have to switch it back. Okay, two x. So here's the picture. X over two equals the sign of Fada. So here's data that makes this X this too, then by the Pythagorean theorem, Um, this side right here equals the square root of four minus x squared. So then the co tangent is adjacent over opposite. So minus 1/4 four minus x squared square root all over X plus c. That's the answer.

Problem Number 85 The artist squared minus one expression suggests a second substitution. So we left. R is equal to say Fator you are which is equal to sec Fator, then fate a e feta and just the limit off the integration so are from one. So the data is equal to zero. And when artists the truth of data is equal to boy over four A boy over three and rewrite the integration. So it's integration from zero provide over three Oh sick a square feet and minus one or 3/2 sec Fator then Fator over sick fainter me fit So this can be simple Find in tow integration from thereto by over three Oh 10/4 feet The Fator eso by a blind formula 93 Ah and letting n is equal to four and is equal to one So the answer for this integration is April toe 10 hour three fatal over three from zero to buy over three minus integration from zero provide over three off 10 squared feta e ah Fator Ah So by applying the reduction Formula One's and gave this time within is equal to 24 This integration value. So the final answer for that will be the square root of three, which is a very for this after some institution from dealing toe by over three and minus. Then so what then? Ah, Fator minus invigoration off the faith from zero to pi over three. And the integration for the theater is equal to Fator. So artist of institution is equal to a square root for three minus square with three plus Overstreet so is equal toe over three.


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