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Problem 8 Medium Difficulty

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

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

Answer

$\frac{1}{4} \ln \left|\frac{e^{x}+2}{e^{x}-2}\right|+C$

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Video Transcript

Okay, This question wants us to evaluate this integral using a table. So to do this, we need to get it into one of our forms. But this really doesn't match anything the table right now, So we should make a clever substitution. So we see that there's a lot of either the excess because we can rewrite this integral as the integral of e to the X over four minus either the X squared because each of the two exes just eat of the X squared. So it seems natural that we should make you saw with you equal the e to the X. So do you is just eat the x d. X. So that gives us the integral of eat of the ex d X is just do you over four minus you squared. So this is close. So one of our forms but our form is you squared minus a squared. So let's factor out a negative one. So this is equal to do you over negative one times negative four plus u squared and this turns into the integral of negative D'You over you squared minus four, which turns into negative times the integral of the U over you squared minus two squared, which is one of our forms. So if we look this up in the integral table, we see that this is keeping our negative sign in front one over two times. A today's too. So it's one over four times the natural log of you minus two over you plus two. So now we can do a little bit more simplification. So we get negative 1/4 Helen of you minus two over you plus two. And remember, by Log Properties we can bring that negative on top of that. Ln so, just flipped the fraction and finally weaken back Substitute. So you was e to the X. So our final answer is 1/4 Ellen of E d. X plus two over each of the X minus two plus C.

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