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Problem 29 Medium Difficulty

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

$ \displaystyle \int \sqrt{e^{2x} - 1}\ dx $

Answer

$\sqrt{e^{2 x}-1}-\cos ^{-1}\left(e^{-x}\right)+C$

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Video Transcript

Okay, so this question wants us to integrate this function. So to do that, we need to transform it into one of the forms we can see in our table. So to do this, we always like to see a U squared insider square roots. So let's pick you to be equal to eat of the ex. So that means that do you is equal to eat of the ex DX. But we don't have that anywhere. So we need to divide by e to the X. So we get do you overeat of the ex is equal to D X So now we can transform. This is the integral of square root eat of the two x you squared minus one divided by archangel factor of e to the X Do you? But we can't keep this eat of the X in that form We need to change the u world now So either the axe is equal to you Then X equals the natural law give you so we can rewrite. This is the integral the square root of you squared minus one over e to the Ellen of you. Do you? Which is equal to well, he's to cancel, and this is an integral we can compute. We just look at the following formula from our table and in this case, a equals one. So that makes are integral square root. You squared minus one minus co sign in verse a over absolute belly of you plus c them plugging back in. We'll get our final answer of each of the two X minus one minus co. Sign in verse of one over absolute value of E to the X, which is just eat of the ex because either the exes always positive plus c giving us our final answer.

University of Michigan - Ann Arbor
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