use-the-table-of-integrals-on-reference-pages-6-10-to-evaluate-the-integral-displaystyle-int-fracx4-

Question

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

Foster Wisusik · Accepted Answer

Okay, so this question wants us to find another anti driven. So to do so, we&#x27;re gonna want to put this into the form of our table into gross. So we usually like to see some sort of a you squared and our denominator So we want you squared to be extra the 10th and that implies that you has to be X to the fifth. So d&#x27;you is five x to the fourth d x, but we only have one x to the fourth. So that means that d you will for five is extra the fourth DX. So now we can substitute So we have the integral of d U Over pulling the 1/5 in front the square roots of you squared minus two. And now if we look, here&#x27;s what are integral Table tells us so In this case, a squared equals two. So that means that our anti derivative is 1/5 from our general factor times Ln of you plus square root, you squared minus hey squared which is two plus c. And now we can simplify this as 1/5 time&#x27;s the natural log of Well, now we just gotta back substitute for you, which we said was X to the fifth X to the fifth plus square roots of you squared, which is X to the 10th minus a squared, which is two plus c. So a relatively simple looking anti derivative for a complicated in secret.

Problem

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

Problem 31 Medium Difficulty

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

$ \displaystyle \int \frac{x^4\ dx}{\sqrt{x^{10} - 2}} $

Answer

$\frac{1}{5} \ln \left|x^{5}+\sqrt{x^{10}-2}\right|+C$

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