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Solve the given problems by integration.The power used by a robotic drilling machine is given by $p=3 \int \frac{\sin \pi t}{2+\cos \pi t} d t,$ where $t$ is the time. Find $p=f(t)$.

Calculus 1 / AB

Chapter 28

Methods of Integration

Section 2

The Basic Logarithmic Form

Integrals

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So in this problem, we're asked that this integral from negative by train to pie, every thready of Stein squared of team with respect with tea. So my first approach to this problem would be to use a reverse chain rule. So we have the girl a function and with argument with it's another function G and argument times the derivative up G is. So the way we do that is we just into great treating g f X as a variable. In this case, we would have em of X is ex cleared and G f X equal to sign of X, so that would make up this part. But we don't really have the derivative of G effects, so we can't use it. So, um, that's okay. There's another thing we can do with this son of ex, and that isthe the double angle double angle farm. So we have this It we have this in the inter ground. So using this formula work going to self for sine squared t and the expression we get that Oregon around substitute here and hopefully that will be easier to deal with. All right, so first things first, um, we want this turn to debate itself. Soon this one went over to the other side. And we do that by subtracting one from this side. And if we do it on this, simply do it for the other side of the creation. So we have to co sign a two times T minus one. Now, this cancels with this and we have on the other side of the oppression two times signs. Where of teeth. Now we want to sign a sine squared to be by itself. We don't know this girl efficient. So we're going to do it. It's divide this side by two now, a days to cancel. But if we don't divide by two here, we need to divide by two for here also. So, um so now we came right explicitly. What sign? Square of T, Mieze. And that isthe this expression. I'm going to write it out to terms. Yes, that's going to be easier to integrate. So first term we have ISS. Huh? One. We have a Oh. Oh, that's it. Okay. Thiss minus sign stays here. Which means this too is here. Which means this to a cell cynic. Okay, Now, um, we are going to write out this expression. So this is equal to thiss minus. I'm going to put it in the top part with the co sign of two tea. This? Yes. Thiss positive. The negative is over here. And this since we have one of her too. Both of them are negative, which means they cancel each other out. And this is positive. So I'm going to write positive terms first, minus Hussein that Tootie Oh, weren't you all right? So, um, this is what we have. That's her inta gram, which means, which means that we can substituted with this expression. All right, so, uh, that's our first step over here. Three. Since we're not integrating just yet, we need to keep this, uh, integration sign. We have minus hoops minus part three and up to a pipe over three. All right. Tonight, instead of sine squared, we have co sign a to Teke over to you minus. Let's go back. One square one. Every two positive terms first, minus tshosane Tootie Over two already. Now we can, but this isthe with respect, Teo t. So now we can start integrating. This's a constant, which means we're going to treat this as a cool, efficient X razed to this year and this We are actually going to use another. We're going to use a reverse change in this case. CO sign it is okay co sign We'LL be out of eggs. Thank you. Case I can't Who signed a fix and J f x walking contempt sti So we have we since we define ji of X to be too t keep private X is too. So we have, uh, this expression get rid of. That's because we're not using it anymore. Okay, So you think, Cass, you put a Geoffrey can, uh, that's whole expression. Okay. ISS the integral of co sign, uh, X but X instead of X. We have chief excess and argument, so they have to t. And this is times the diameter, which is too. And in this case, it's not with respected access with respected t. So then the ship PT Nishiki. Uh, so comparing that to let we to have we don't have to, uh, but we can fix that. Bye. A little too plain to buy here. Which means if we multiply by here, we also have to divide by two. All right, so now it's ready to go. It's set up to use to use a reverse change rule. And this will be cool, too. No. So then we No, actually, that's stood over here on. Yeah. So for this, we are using the powerful. Her system different. Had a dirt of the integral of X rays to some power and different from minus one. Um, well, respective eggs. This is equal to X and plus fun and divided by this new park, plus c a bit indefinite. No, we don't need to worry about this, because in this case, it is next. Here. Uh, we're gonna integrate twelve. Righto. So center integrating. We're not writing this part. And we have one over two. We keep that because it is a constant, a coefficient and x zero. So now we just need to out of one here and divide by one. But we'd already x to the one is just X and tense. One. We don't need to write it orchestrated. Well, it's a good completion, so we're not using that anymore. All right, so now for this. Jake, your part, we have this minus. We'd keep it there um, this will break This will be a constant. So we just write. Keep this, uh, one over four and co sign, uh, be a co sign. Uh, duty won't apply by two. Stone out this one. Oops. Uh, not that one, either. Okay, so and this case, we are just going to integrate this f in this case co sign creating TF X. That's two of us treating. Give X. That's this. Ask a variable. And this one, we don't need it. We just needed to be thick, so this will be equal to sign up. She x t. All right. Plus a constant. If it's under find again, we're not using that in this case. So this becomes sign, uh, Tuki. And then this is a constant that we kept hitter. All right. And then this is against senses to say. Definite. Integral. We have that tea. This expressionist evaluated from t equal to Linus. Pipe over three. Too high. Over three. All right, So, uh, nuts. Clean this up, and then we're gonna evaluate here. Um Okay. So, um, just we don't need to write a thesis. Ones. You think that is all the cleaning up? We can do. So we have X over too. Minus sign. Uh, O sixty T t a tear or two. The sign of two tea? No, over four. And then this is a reading from King equal to minus pi over three a two. Priority three. So now, Angus Flint, this expression, what it means is we're gonna evaluate this function when t equals pi. Oprah. Three. Now we're going to subtract there seem quick, the same function of alligator t equal to Linus pie over so running out of state. So we're going to write it over here. You're gonna start with the equal to pi over. Okay, then we subtract Same expression twenty equal. He wants to minus pirate tree. All right, so it's empty, uh, substituting. So we have tea over to that is tight. Over. Great. Over to two. That means, and that, too, is over here. Minus sign of two t over four. Sign done. Two times p over three. Silly have p with to multiply gratitude over here every three. And cause this is Laura's work. First violation. And then we subtract the same thing. But twenty takes this time. Certainly. Yeah. My foot minus for tea over to have tea so over to we wouldn't put the two over here minus sign. Uh, two times. T. So that would be Yeah. Minus who braided over here. Pi over three. And that's this over more. All right, self, Dysart simplifying this, um, This We're just gonna have to re great. Right. So So the signs phone this sign of Pete over two terms, Pete over three. So let's just read it off. That's rather bitter. Okay, First term iss p. Opie over six minus. And there's a sign evaluated at this angle. ISS Mine is a squared of three over, too. And then this stays here. It's a constant. Um, okay, so now for this one, we have minus and minus, it turns into a positive get over. And it's a six again. This minus, and this minus is positive. Now, when sinus evaluated at this animal, it's equal to Linus square root of three over, too. And then we can forget this for it. So we just put it in there night. So, um, now we're gonna bring together these two expressions that up pipe and these two expressions that have, um, the square root of three. Since I difficult to combine Sepp, these two expressions since you're the same, it's two times people her six. And for these too, we have minus, uh plus the same thing. So again, it is twice that amount. But it's negative. So we have times minus two times square root of three over two times. Um, to make things a little bit, he's here to see I'm going to write this again as two times three. Ah ha! Fewer and going with. So these two twos cancel out in these Tutu's also cancel out. So what are we left with? We have by over three here and square root of three number four. All right, so this isthe this is what this integral it leads to

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The number 2 is also the smallest & first prime number (since every other even number is divisible by two).

If you write pi (to the first two decimal places of 3.14) backwards, in big, block letters it actually reads "PIE".

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