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Problem 28 Medium Difficulty

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

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

Answer

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

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Video Transcript

Okay. This question wants us to integrate this function. Using a table so well, usually like, is having some sort of a you squared in the denominator. So let's have you equal eat of the ex. So that means that do you equals you to the x d x. But we don't have any of those. So do you. Over e to the X equals DX. So no, we can write this as d'you over either the X square roots one minus you squared. But we're still not done because we haven't eat of the ex and we're supposed to be in the u world. So if you is equal to eat of the X, then X is equal to Ellen of you. So now we can rewrite this as the integral of D'You over E to the Ellen of you Square group one minus you squared and we see the deed to the Ellen. You is just you. So we get our final integral of d U. Over you square root one minus you squared. And this is something that we can look up in our table. It's exactly one of the forms. So we get that this is equal to negative times. Ln of absolute value, one plus square root one minus You squared all over you plus C And now we can flip this fraction over and put them to get rid of the negative sign. So we get equals. Ellen of you over one plus square, one minus you squared plus c. And then the last thing we got to d'oh is plug in eat of the ex for you and now we're done.

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