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JH
Numerade Educator

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Problem 29 Medium Difficulty

Evaluate the integral.

$ \displaystyle \int \ln (x + \sqrt{x^2 - 1})\ dx $

Answer

$x \ln \left(x+\sqrt{x^{2}-1}\right)-\sqrt{x^{2}-1}+C$

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Video Transcript

let's use integration. My parts here, let's just take you to be the entire intolerant. So most of our work here will be and finding and then simplifying to you So here will get the derivative of the log and then we'Ll also apply the chain rule here. So on the time we'LL have the multiplied by the derivative of this term here in the denominator So that'LL be one plus And then after simplifying right when we differentiate the radical, we get this So here for our next let's just go ahead and get a common denominator up there in the numerator. So multiply it top and bottom of won by the radical So that will be our numerator. And then we still have our denominator. And here before we simplify anything noticed that these two terms in the parentheses are exactly the same. So we could cancel them and we left over with one over X squared minus one d X. So all that work was just to find our do you And then, since we chose you to be the entire Interbrand we're left over with, DV equals DX, so V equals X. So the next part here will be to replace the original integral with the formula from integration. My parts. So let's go on to the next page and rewrite this formula here using our choice is of you, Andy. And here we see that we'LL need to you as well. So we'LL need just her So going on to the next page we still have you the inner grove e t u And then let's plug in our Eun Bi's Finally. So we get extra times the natural log and then minus in a girl And then we'LL have a X over tyrannical. So we just have one more integral to evaluate. But this animal is simpler than the original, and we could just do another substitution here. Let's take X squared minus one dw over, too. No x t x so that in a girl just becomes one half and then we just have w to the negative one half a t w. So after cancelling that one half with the two that you get from the power rule so we just get X squared minus one to the one half. So finally, let's just rewrite this UV term so x times log again. And then don't forget the minus here in between minus. We just evaluated this inaugural over here using the use of it became this radical. And since we've evaluated all the intervals, let's go ahead and add that Sian and that's your final answer.