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4,4.1 ExercisesEvaluate each of the following integralcosVt-] Vt - 12(3f _ 1)? dtletane 'sec-0 d0cos (S0 )sin(50) d0x(312 -1) dx2x(x? 1) dx...

Question

4,4.1 ExercisesEvaluate each of the following integralcosVt-] Vt - 12(3f _ 1)? dtletane 'sec-0 d0cos (S0 )sin(50) d0x(312 -1) dx2x(x? 1) dx

4,4.1 Exercises Evaluate each of the following integral cosVt-] Vt - 1 2(3f _ 1)? dtl etane 'sec-0 d0 cos (S0 )sin(50) d0 x(312 -1) dx 2x(x? 1) dx



Answers

Evaluate the integrals in Exercises $1-22$ .
$$\int 7 \cos ^{7} t d t$$

All right. So now we have a integral of co sign to the Q Coast on cue the four x he X and we know that, um, we want to get this into a form where we can use u substitution toe to integrate this so we know that we have co. Thank you. We can break that down into co seven squared for X times co sign of forex DX. And now we have our our our our piece needed, um, as are inside. See what? How to get this use of. We have a co signer for X DX, which, um, we know that if we have a sign of sign of forex, as are you, it will give us a d. U of ah portion of this so we can go ahead and use a trick it and try identities to turn this into a one minus co signed a one minus sine squared forex Times co side of forex the X and you can now use our use of technic of used vehicle to sign a forex. Sorry about the blank's and giving us the U of four co sign forex. The axe. So then now we get a 1/4 in the front in agro of one minus. You squared you and we can go ahead and integrate this three of us 1/4 times U minus. You cubed over three. And since this is a, um, substitution problem, we're not gonna write c just yet until the end. So then we end up with 1/4 sign of four x minus. Sign que the four x all divided by three. And you can distribute, um, together the simpler answer. But I'm just gonna leave it with plus C, and that's our final answer.

All right, So we have the Inter goal sign Cube decks co signing cute decks DX. I'm gonna go and solve this so we can choose either sign or co sign. Since they're both cube to be are you sub? And we're gonna want to keep signs. It makes it easier since you don't have to introduce another negative. So we're gonna go ahead and turn one of these coastlines into signs leaving us with just coastline DX. So we have integral of sign que deck surge in the first part and then we take on a coastline squared X and then a co sign X. Yeah, No, we can use our identity to give us one minus science. Codex Time to co sign indeed, cause the next DX And then we can go ahead and consolidate all of this, leaving us sign Cubed X might assigned to the fifth X Times co sign next DX and now we can pull out. Are you serve, you assign X and then the U is co sign X DX. Now we can simply just turn is signed into use giving us the integral from just integral of you cubed minus you to the fifth Do You Were you for the fourth over? Four minus you to sixth over six, Which in this case, is signs the fourth X over four minus signs. Six. X over six plus C since it's an indefinite integral.

All rights, we're going to the integral of co sign squared X, the X. And now we want to go ahead and think about how we can solve this with identities. So we can't simply solve this through any any means. As before, we bring signs and coastlines because we have no signs of coastlines to do that with. So we had to think about what identities we can get a coastline square from. So, um, there is a pretty simple identity, Um, which is a double angle formula of So you get co signed two x is co sine squared X minus sine squared x. Okay, then you think about how could we use this to get this co sign term by itself? So we know that we can use our identity, uh, to switch this to co signs to give us co sign two X is co sine squared X minus and then sign is simply one minus co sign squared X. So then we have this identity that gives us co signed two X is to co sign squared X minus one. And if we go ahead and, um, if I rearrange the terms, we end up with our identity code sine squared X is equal to, um one plus co sign two x, all invented by two. And now we can go ahead. Insert this because these are two terms that air easily Integrate, integrated, integrate herbal So we can go ahead and do this. It turns into the integral of one plus co signed two x over to DX pulling out of 1/2. We end up with 1/2 times the integral of one plus co sign two x, the X and now we can just get that This is 1/2 times x plus, uh, sign two x divided by two plus c So this is our final answer.

When I first laid eyes on this problem just a few short minutes ago, it almost brought me to tears. And not because it was too challenging, but because it's so beautiful. And I want you to see why that the only thing we have to do they simplify. Let me show you why, If we rewrite this problem to simply cancel out the bottom of the top, you have something like this. The Inter girl. All right? Yes. The in general of one over t d T. Now why can I do this? Well, as you can see, tangent of the tension times the coastline of the inverse co sign of tea actually simplifies cancels out with the square root of one minus t square to just one over t d t. And this is the answer. If we go ahead and we take Oh, now we can just take, uh we just know that this is a standard in the girl in the interval of one over teeth is simply the natural longer than Morrell. End of tea. So, Allen of the absolute value of teeth plus C, that's our answer. Next thing we're gonna do is find the corresponding definitely room were given from 1/4 to 1. If we go ahead and do that, we have to evaluate the integrative at the limits and subtract, which I'll go ahead and do appear well. Actually, it's not next page. So this from three ln of the absolute value of tea, given from the interval of 1/2 of shooting from 1/4 to 1/2 this is equal to ln 1/2 the natural on rhythm, 1/2 minus the natural rhythm of 1/4. And this is equal to Allen of To the natural log rhythm of two. And that's our answer, Alan of two.


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