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Solve the given problems by integration.In the analysis of the intensity of light from a certain source, the equation $I=A \int_{-a / 2}^{a / 2} \cos ^{2}[b \pi(c-x)] d x$ is used. Here, $A, a, b,$ and $c$ are constants. Evaluate this integral. (The simplification is quite lengthy.)

$\frac{a A}{2}+\frac{A}{2 b \pi} \sin a b \pi \cos 2 b c \pi$

Calculus 1 / AB

Chapter 28

Methods of Integration

Section 5

Other Trigonometric Forms

Integrals

Harvey Mudd College

Baylor University

University of Nottingham

Boston College

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So that is Have okay. Tan squared, ex seek. And okay, uh, seeking to base the ex a cave. Okay, so we have this one. Uh, is he? This is techniques off integration techniques off integration means that there are a lot of things that you can try that is not gonna work. You have to combine different techniques in your head and try to see which one is gonna fit. Okay, Uh, right off the top of my head. What I can see here is that I have to convert this one. Okay, because, uh, 10 10 squared is difficult to integrate, but I can convert it to seek in squared. OK, so we know that there's a relationship here. Tan squared X plus one is seeking squared exodus in trigonometry Gate. So, uh, here I knew that I can write, uh, this tan squared to convert it to seeking squared. OK, tens. We can be written as sequence. Where X minus one from here. Tens. Where this second squared, X minus one. Okay. And in this is That's right. Here came. Now I'm getting a foil. So this one is gonna be seeking to cute x minus seeking the base. Okay. Yes, that is the foil in. Okay, I just distributed the seeking to these two terms. OK, then I have this one. Now I got to deal with them separately. Okay, So this one is seeking cute X d s K minus. Uh, because integration is a linear operator, so you can distribute it right away like that. Okay, Now, let's consider this one, because that that one is that hardest, this one is easier. Okay, so let's consider the harder part first. So this one becomes, uh you know this second squared X here, the X. Okay. I'm gonna write it as seeking. Swear x, second X, the exit gate. And when a convergence one into tangent. Uh, Okay. So Okay. So, uh, there are many ways you can do this one. You can change this one. It's a tangent to try to make you a substitution, but you're gonna struggle. Okay. Uh, but you can use something called a reduction formula and a reduction formula. Okay. Eso When you do that, you're gonna get okay. The reduction formula is, um you have this and then seek and end of X'd extricate reduction formula is seeking n minus one ace and then over in minus one. Um, sign eggs. Sign eggs here. Okay, then. Plus en minus two over N minus one. Integral. Uh, second, an minus two ace DS. This is a reduction formula. So when you use that one, you can use it. Break this one here into, uh, a into this piece right here so that you can integrated as easily as possible. So if we use the reduction formula here, you see this? And here's three. Okay. Is three in our case. Okay, So he's gonna be seeking sweet X over too. Okay, Because in history and he was gonna be signed. Exit gate, then close en minus two is one an n minus one is too. Okay. And they have second and minus two. So this is just seeking to X, because industry through monies to his one. Okay, so the reduction of formulas made it very easier for us. This one is easily in a global. Okay, it is the natural log. Okay, natural log off seeking explodes. Tangent X. And then this one right here is you know, weaken. Simplify. This one seeking is one over co sign. OK, so we can just take one seeking to wait. And then this is gonna be signed over. Co sign right here, because seeking is one of the co size this is gonna be signing because I, which is tangent of excess over two. Okay, so is a bit messy, but, uh, this one also works. Okay, so we have this one here and then close that. Okay, so that is the interval off, this one using the reduction formula. Okay, Now I'm gonna clean it so that the world becomes in the night a little nicer. Okay, so that is the reduction form that you can search about it. Production informants for sequins. So that is the integration. And then now we have this one to integrate. Okay, this 12 it's easier for gay. This one is also, uh, in a group of this one seeking X. The X is natural log. Okay. Seeking to eggs, Close tangent X again. So I would have to do now is just, uh combine it. OK, so, like, we did the last tutorial, call this equation one and Clovis equation too. Okay. And do that, and you do equation to mine equation. one minus Equation two. Okay. And do that equation one minus equation to then you have natural log off. Second. Thanks. Most Andean inx minus half off the same thing A gay. So 1/2 of the same thing is gonna give you 1/2 natural oak seeking to expose tangent, Extricate, then minus this guy right here. So it's negative. Seeking to eggs Change into X. Also 1/2. Okay. And then close. See? So, uh, it is not so pretty, but it is effective gain. So, uh, okay. We were supposed to do Yeah, one minus two. So is this one. And then this is too. It was supposed to do to minus one. Okay, Because it is this one minus this one. So supposed to be two minus while not one minus two k. So, uh, less less. Make that correction. So is to minus one. Okay, so to minus one. So we're taking this one away from, uh, this one right here. Okay. Because this is their relation. It is this one minus this 1 90 attari around. Okay, Said, let's make it better. Okay. So, uh, when we take this one away from this one Then you can see that the sequins 1st 1 is gonna be 1/2. Okay, The 1st 1 is gonna be 1/2 right here. One have seeking eggs. Tangent X. That is the 1st 1 Then we take this one away from this one. Okay, so when you have half of something, you take one away from that. You have negative half of it left. Okay. Seemed seeking to x close a tangent X C before that becomes an answer to

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