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Numerade Educator

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Problem 25 Easy Difficulty

Use the Table of Integrals on Reference Pages 6-10 to evaluate the integral.

$ \displaystyle \int \frac{\sqrt{4 + (\ln x)^2}}{x}\ dx $

Answer

$$\frac{1}{2}(\ln x) \sqrt{4+(\ln x)^{2}}+2 \ln [\ln x+\sqrt{4+(\ln x)^{2}}]+C$$

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Video Transcript

Okay, This question gives us this end girl to evaluate. So to do this, we're gonna want to put it into a nice form that we can look up in the table so that Ellen axes what's causing problems, So let you equal Ln of eggs. So that means that do you pickles one over x d x, which conveniently enough we have in our integral. So this means are integral becomes the integral of square root four plus u squared. Do you? And we can fix this integral sign a little bit. Okay, so now from there, there's a very familiar in circle based on our table. So we see the following formula, and in this case, a A's equal to his two squared is for so are integral becomes you over too square root four plus u squared plus four over too Ellen of you plus square root, a squared plus u squared plus c. But we could replace this a squared with before and now all we have to do is we can make one simplification with this four over too. And we can just plug in Alan of X in for you. So we get Ln of X over too Times Square Root four plus Ellen of X squared plus to Ln both Ellen X plus Square root for Plus Ellen of X Squared plus c. So we got some nested longer of them's here, but being extra careful with everything, this is our final anti derivative.