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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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Video Transcript

Okay, so this question wants us to find another anti driven. So to do so, we'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'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'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'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.