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JH
Numerade Educator

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Problem 10 Easy Difficulty

Evaluate the integral.

$ \displaystyle \int \frac{\cos (\frac{1}{x})}{x^3}\ dx $

Answer

$-\frac{1}{x} \sin \left(\frac{1}{x}\right)-\cos \left(\frac{1}{x}\right)+C$

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Video Transcript

let's start by taking the U substitution. So looking inside the co sign, we see a one over X. So let's just try that, then using the powerful for derivatives. So here it looks like we can rewrite this using you. So it's plot of minus sign here because of this one from the use of So by writing negative, do you? We're getting one over X squared and e x so that gives us this and X squared on the bottom. So on top. So here, let's write. This is CO sign you. This is giving me this over here and by writing negative you deal. That's giving me everything else. The negative do you is giving you two of the ex is on the bottom and then multiplying by this extra you is just giving us another one over x. So that gives us the one over X cute. So this is our general. So we have negative inaugural of you times co sign you So for this new in a girl here. So it's got a little messy of here. Let me come and rewrite this. That's right. This is you co sign you. Do you now let's set ups the integration by parts for this one. So let me not use the letter you since we were to use that up here and it's in our general. What's his w equals you? T w deal TV is co sign So we assign you so using integration my parts. So remember the formula. There's UV minus integral ggo. In our case, let me go ahead and replace that with a w there. Since that's what was that we're using So negative. Then we have you time sign you minus in a girl sign you That's B and then d w which is do you? So now we can go ahead and distribute the minus sign. So that's a negative. You signed you plus integral of sign. So that's negative. You sign you minus co sign you. And now the frigate at your constancy. Now we've evaluated the intern group with the last step here will be to come up to this u substitution so that you can replace you in terms of X. So this last step here is just to do that, replace you. So you is one over X sign one over X and then we have minus co sign one of Rex and then plus C and that's your final answer