use-the-table-of-integrals-on-reference-pages-6-10-to-evaluate-the-integral-displaystyle-int-fracs-3

Question

Use the Table of Integrals on Reference Pages 6-10 to evaluate the integral.
 
$ \displaystyle \int \frac{\sqrt{4 + (\ln x)^2}}{x}\ dx $

Foster Wisusik · Accepted Answer

Okay, This question gives us this end girl to evaluate. So to do this, we&#x27;re gonna want to put it into a nice form that we can look up in the table so that Ellen axes what&#x27;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&#x27;s a very familiar in circle based on our table. So we see the following formula, and in this case, a A&#x27;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&#x27;s here, but being extra careful with everything, this is our final anti derivative.

Problem

Use the Table of Integrals on Reference Pages 6-1…

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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