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JH
Numerade Educator

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Problem 22 Easy Difficulty

Evaluate the integral.

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

Answer

$\int \frac{\ln x}{x \sqrt{1+(\ln x)^{2}}} d x=\sqrt{(\ln x)^{2}+1}+c$

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Video Transcript

Let's ease EU substitution here. Let's take you to be the natural log Then do you run and we see the one over X B c x here in the denominator and we see DX over there. So that corresponds to these two terms over here on the right. So after this use of we can rewrite this, we have a you appear less Ellen of X. And then do you then in the denominator, I have to rewrite this radical is one plus you square. So that's our new integral. And then here we could do another use a bliss to w equals one plus use. Where, then here, If you do dw and then divide by two that corresponds to this numerator here you do you so one half. Then we have dw up top and then square root of w on the bottom. So just use the power rule here. So one half then w to the one half power Then divide by one half at our constant of integration. See, cancel those one half and then replaced w and in terms of you So that becomes where one plus you square and then finally use the original U substitution u equals Ellen X to rewrite this. Not this square root, plus our constancy, and that's your final answer.