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Find the arc length of the curve $y=\ln x$ from $x=1$ to $x=2 .$

$L=\sqrt{5}-\sqrt{2}+\ln{\frac{\sqrt5-1}{2\sqrt2-2}}$

Calculus 1 / AB

Calculus 2 / BC

Chapter 7

PRINCIPLES OF INTEGRAL EVALUATION

Section 4

Trigonometric Substitutions

Integrals

Integration

Integration Techniques

Trig Integrals

Trig Substitution

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mhm. Okay, so we want to find your length of the function. Why equals natural long of x from two points and recall that York length is equal to the following inclusion. So l is equal to be integral from a to B of the square root of one. Close your derivative squared. All right, so my crime is equal to one over X. So what you get is the integral from one to of one plus one over X square DX. All right, so if we let one over x squared equal tangent see, the this means that we're sorry. One of our expert tangents. Great Tita, This means that Q X is equal to call these action. They forgot the step. So what we're going to do is write that one plus one over X squared is equal to x square plus one over x queary. So if we put all that under square roots, that is equal Thio X squared plus one sweet route over X. And this is an integral you've seen before. Now, if we let x Equal Tangent Dina the X equal second squared theater theater. This employee aims expert plus one. Well, the seeking squared so the square root will get rid of the power. So we're going to get is the integral from 1 to 2 of seeking theater over 10 Jin theater multiplying place You can squared theater, Do you say that? All right. And so again, this 10 Geant can be brought out as co tien Geant, which is co signing over sign. And so that second right there is going to cancel with co sign. So you're going to get you can square data. All right, so now we have the integral off Seacon Square casita saying theater 12 you see that? Okay, so let's change that. Seeking to one plus t engine square, theater or sign. See the So let's just evaluate the cynical That is one of her sign, which is Corsica, Tina. Plus tension squared Is science squared over cosine squared. So we get rid of one power of selling over cosine squared. Okay. And so what we're working with here is the integral of Corsica. I'm gonna move way down over here to the left. So the integral with Cosi Kinte is the natural long, uh, cose Eakins. See that when this contingent theater and then the integral of sine. Okay, so sine theta or co sign is tangent, And then we have times one of her sewing again, which is seeking. So this is seeking tangent, integral of seeking tangent. Just second. Okay, so that's the general integral. We just you don't need the general Integral. Okay? And so long time ago, we said that the X was equal attention. See that? And so when you get back to ex we see this is opposite over adjacent. This is Tina. This is X squared plus one soco. Second is I got newest her opposite. So x squared plus one or X Coty engine is simply one over tangent. So, uh, so that's gonna be one over X. This is already HRIC, so I'm just going to extend that, and, uh, she consider we knew a long time ago was expert plus one. Alright, so we evaluate that from one to Yeah, So you're going to get this the natural long of five minus one over to plus the square to five. Then you're going to get the natural log of to screw Joe minus one over one plus Screwed of to. Okay, so this is the same thing is seeing natural log off so we can do here. Innocence thes natural lungs are subtracted is we can divide them. So square root of five minus one or two divided by this. So too square, too minus two. All right. And then plus the screw five plus the screwed too. And I believe you're just about done. Actually. Just realized the a little ever hear, so that should be negative. And now we're done.

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