This Question: 5 pts24 of 25 (0 complete)Find the derivativesin 3t dt a. by evaluating the integral and differentiating the result: b. by differentiating the integr...

Evaluate the integral using (a) the given integration limits and (b) the limits obtained by trigonometric substitution. $$ \int_{0}^{\sqrt{3} / 2} \frac{1}{\left(1-t^{2}\right)^{5 / 2}} d t $$

This problem is from Chapter seven section to Problem number five in the book Capitalists Early Transcendental Sze eighth Edition by James Story Here we have ah, indefinite Integral science of the fifth power of two t co sign squared of two tea. So here, since the power of sign is odd, the first thing we do is rewrite this by pulling out a factor of signs to have signed in the fourth Coastline Square. And let's put that extra factor of sign over here at the end. Chris, observe here in the inner grand that the science of the fourth power. So it's going to decide and rewrite this so we can rewrite sign square science of the four power as Science Square Square. Then we could apply, uh, put that in identity to rewrite sine squared is one minus coastlines where, and we have to square this entire expression and finally weaken. Evaluate this latest expression two co sign squared to team plus coastlines in the fourth power two teams, so replacing science of the forth. With this newest expression, we have the integral one. Minus two co signed square plus coast under the fourth Power Time's Cose identity to co sign squared to tease and sign off duty. So here it looks like we can apply u substitution here we should take you two be co sign opportunity, which implies to you is negative, too. Sign two t d t. That's using the chain rule and here because we don't have a negative to in front of the sign and the original under grand, we should make up for this, but by multiplying by a negative one half in front of the to you to give us exactly what we want. Sign two tea Titi Just as appears an immigrant. So now let's rewrite this integral using our U substitution. So first, let's pull off this negative on half outside the integral level one minus to use Claire. Plus, you know, the fourth then co sign squared is just use Claire and then to you. So now let's just leave this negative one half outside for now. And what's the stripy tissue squared inside the apprentices. So we have a use clear minus to you to the fourth power. Plus, you know the city's power to you to evaluate the center girl. We should supply the power rule three times. So we have you to the third power over three minus to you to the fifth over five. Plus, you're the seven over seven, plus our constancy of integration. So here we have two steps left will replace you with co sign of two tea, and we could multiply by this negative one have so we get a negative cool sign. Cube's Tootie over six. Here we have a plus coastline to the fifth hour over five. Those two's cancel. Now we have a minus co sign to the seventh hour, all divided by fourteen and closer constancy, and that's our answer.

Okay, We're going to integrate. Sign Tiu. Who? Us? Close t zero said pie over four. That's DT. Um, yes. Okay. So, using the fundamental theorem of calculus, we know that this is equal to the anti derivative of that expression. Evaluated at by before minus the anti derivative, the same expression evaluated zero. And so the anti derivative of sign is negative. Co sign T, and the antidote of coast is positive. Sign. So positive sign. T begin zero and pirate for take the difference. So negative Coasts of pi over four, plus sign of Aye. Or four minus brackets. Here. Negative coasts. Uh, zero poor us. Sign of zero. Okay, so you just seem fine. Now, what is coast of pie before? Musically, hand trick. And I know that it's rude to over two. So negative route to over two and sign a part for the same. So plus route, too. Over too. Minus coast of zero is one. So that's negative. One. They never signed there. And then sign of zero is zero. So plus zero. So these cancel out to zero. And now this negative distributes through. So the whole answer here is one. Okay,

Pushing. Would you call that? It went into go off nests I I'm the new about and the new Then we should get a cogent uba and this one over and this Qantas e and now in the question were given into Go go side Judy times this sign to the power five that would be isn't for us to see And now I'm going to Judy's only with you. So you go to the side duty and then the new eco to the ghost side ut and buy the General Winnetou Temps due to the front and then Thames DT So it means that that did the echo to the renewal number two go side to and fro. We can read this integral in terms of the deal now. So close. I do. He stayed the same times a year. About five. Now now, Did you? Could you Did you remember to go Sai Judy? We see Then we can consider this and this and a number of joyous a constant we can bring our side. Therefore, we get a coach half integral on the year above five year now by formula, we're gonna cut your half you both six almost six plus c, but new ego to decide ut. So we have one of the trial, and inside you, the power six Bless.