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Evaluate the integrals$$\int_{1}^{32} x^{-6 / 5} d x$$

$\frac{5}{2}$

Calculus 1 / AB

Chapter 5

Integrals

Section 4

The Fundamental Theorem of Calculus

Missouri State University

Oregon State University

Harvey Mudd College

University of Michigan - Ann Arbor

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soon in this problem, we are asked to evaluate this integral. And so the first thing that we notice spotted it's that it's an integral off the power eggs and were derived integrating with respect to X. So we the first thing that we're going to recall this tower pour into girl. So for integrating X lay still power and where in has to be different from one. Uh, this tentacle will be ex raced and plus one and we divide bye and plus one to this is a indefinite And the growth we are, eh? A constant in this case, we don't really need it. But so in our case and ISS minus sixty five. So first thing that we want to do to integrate this pendant in plus one. So since this is a fraction of going to do it separately, yeah, and plus one isthe over six from seven hundred six over five plus one. But instead of range one since we want to be able to add a two two numbers were one of them is a fraction they have to have the same the number denominator. So we want something with sign on the bottom Since we're talking about one, we want same number asked in a greater and greater. So instead of writing one working to write five five. Still, this is equal to five days right here and now top arguably. Haps my six plus fine. And that is people too. Minus one over. So this isthe class. Fine. Which is what we're gonna do. Okay, uh, look. And here and here are integrated. So and this first up, we don't need this practice. We're already integrating, and we have ex place to n plus one, which we've figured out this minus one over pi Delighted by the scene number Mendes one over. And this expression, it's going to be when x equal one and up to thirty two. So before, without practice, huh? Function in thes and points of the integral. I'm home to reeling to this so that it's easier to work with. So we have, uh, this minus saying we'll keep it here. We could put it instead over here instead of here. And this five since it's in the numerator after the narrator, we can bring it back here to bring it up floor here. All right, so we can rewrite it like minus five eggs. Fruit base, too. Mice one. Over and again. We don't lose this X equal to one, two, thirty two. All right, so now, before we do this next part, which issue seemed the fundamental Purim of calculus where we evaluate this when ex house and one x thirty two minus the same expression on except one we are going to I remember some of that fact. So here we have. When we raise one to any power, get one and thirty two, we can write that us to the park soon. I am going to first one X equals to this first part is when X equals thirty two, which we're going to raid us two to the fifth power and then we're going to use the same expression when X equals one. All right, so let's move this downtown. So we have Manus fire. That's a constant Tim Specs, just two to the five. And this text is going to be raised to one. Men's went over and bring to this is the expression on X equals thirty two and three Subtract the same expression on X equals y. So the constant does not change eggs. This one and the power. All right, so you know, it's all constants. We can no work on getting this a little bit. One similar simplifying this right? So we have. And this power, it's five times who won. Minus whatever fight the chain. Since this is denominator, the numerator and denominator thes cancel out. So we're left with two races to the minus one, which is people to one half. And in this case, since we have one race to some power, this is going to be equal to just so this simplifies too. Minus five times one half between kin, Right? Like this. And this place no, outside this minus and minus turned into positive. And this is five times one. The chains. It's just five. Now again, tow up to numbers we have. This is a fraction and a hostin a mayor too. So we're going to put it to here. And since we do it on the bottom of this number, we're goingto have to do it the same. No matter. So think of this five as five over one. That's what it is. And then we're just multiplying by to both the numerator generator. Right? So now we can actually at these numbers till you have minus O on the bottom. We have to. That does not change. And we have minutes five plus times two. And that's equal to five. Number two seven. That isthe our answer when we evaluate this tibia.

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