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Problem 10 Medium Difficulty

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

$ \displaystyle \int \frac{\sqrt{2y^2 - 3}}{y^2}\ dy $

Answer

$-\frac{\sqrt{2 y^{2}-3}}{y}+\sqrt{2} \ln \left|\sqrt{2} y+\sqrt{2 y^{2}-3}\right|+C$

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Video Transcript

Okay. This question wants us to integrate this function using our table of any girls. So if we look in the back, we see that this most closely resembles the following formula. So in order to evaluate are integral. We want to put it to look like this form. So to do that, we need a use squared in our square root. So let you squared equal to y squared. And that means that you cols the square root of two times Why? Or why equals you divided by squared of two. And this also means that why squared equals you squared over two And now we confined Our general factor of do you equals spruced sooty Why or d y equals do you over route too. So now we have everything we need to transform this integral so we could start plugging things in. So now we have the integral with square root of while to y squared Is she squared minus three, divided by Why squared? Wish we said is you squared over two times Do you? Which is to d y? So we need to you over too to Sorry and this simplifies to two over route too. Times the integral square root you squared minus three. Over you squared, do you? And this is exactly the form in our table. So we get to over squared of two times square, reach of the negative sign in front. You squared minus a squared, which is three. What about you? Plus Ellen of we should keep absolute values here. You plus the square root of you squared minus a squared. Plus C Now we can back substitute. So you squared is too. I squared and the U is route to why Plus natural log of you is again route to why plus the square root of two y squared minus three plus c And then we could just do a final distribution here. So we get negative square root to y squared minus three over why plus well to over route, too. Times the natural log of route to why plus square root of two y squared minus three plus c. And this is our final answer

University of Michigan - Ann Arbor
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