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Integrate each of the given functions.$$\int 3 \cos ^{3} T d T$$

$3 \sin T-u=\sin ^{3} T$

Calculus 1 / AB

Chapter 28

Methods of Integration

Section 5

Other Trigonometric Forms

Integrals

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so high said this question we're gonna be solving for each ago off the coast of 80 Cube DT. So the first thing we're gonna do is we're going Teoh, rewrite this as the coasts of 80 squared times the coast of a t on we have our DT on. So there's an equation that states that the coasts of X squared plus the stein of X squared is equal to one. So the coasts of X squared is equal to one minus the sign of X squared. We're just gonna plug that in. So we're going to have the integral of the one minus the sign of a T squared times the coast of 80 DT And from here, which is going to distribute our coats of 80. So we're going to have the coasts of 80 minus the side of 80 squared times the coasts of 80 and we have our GT Uh, now from here, we're just going to separate this into trying different inter girls. We're gonna have the integral of of a T d. T minus the integral of the sign of 80 squared M c coast uh, a t e t. I will be right, that a little better. So sign of a T squared coasts 80 DT. Okay. And from here, we're going to do is we're going to do to use substitution. So our first one we're gonna have a U is equal to the sign. It's just sign. Sorry, not sends word of a t. And then we're gonna take the derivative bow ties with the EU is going to be equal to a Khost 80 that she shooting chain rule E t on. So from here gonna divide both sides by a coast eighties. We're gonna have d you over a coasts A T equals DT. And over here I'm gonna call it actually a the substitution because I don't want to get us confused between the U and like having to use. So we're gonna say V is equal to 80 on dso weren't driven of both sides to d V is equal to a d. T and organ divide both sides by exit. My a d d over a is equal to d t. And now we're gonna plug these variables end to our equations. So we're gonna now have who of V because we're gonna plug in RV for 80 and we're gonna plug in our d v over a for R d t. So we're gonna have TV over a on We're gonna subtract now and we're going to have the integral of you were gonna plug in Are you for our sign A T squared? Um, And now we're gonna have times our coast a T and we're gonna plug in D you over a coast 80 for our DT. So we're gonna have d'you over a coast attempts Khost a T. And from here, we can simplify. We can cross out our coast 80 and we can take out our one over a suitable to have in temporal of coast of the D V minus. I'm going to take out or a as well, one over a times the integral of you square d you and we're gonna use thean equation that states that the equal of X to the n is equal to X. I want to do this in a different color. Um, to the integral of X to the n is equal to X to the end, plus one over and plus one. And so from here what we're going to do is ah, we can solve for are integral. So we're gonna have one over a times the integral of the coast of the Devi, which is just going to be equal. Teoh, sign on. This is gonna be in minus one over. Okay, times. Um, are you squared? So D view. So that's gonna be you to the third or you cubed over three on. We have plus R see, I just have one c for both. That's all we really need. We would be simplifying to that anyway. And so we have Our V is going to be equal to 80 and we have Are you is equal to sign 80 on from here. We're just going to plug in. So we're going tohave one over a sign of a T minus one over a times the sign of a T cubed plus C and this is over three. And so this can simplify. Be sign of 80 over a minus the sign of 80 cube over three. Any plus C and that is going to be our final answer

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