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Problem 21 Easy Difficulty

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

$ \displaystyle \int \frac{e^x}{3 - e^{2x}}\ dx $

Answer

$\frac{1}{2 \sqrt{3}} \ln \left|\frac{e^{x}+\sqrt{3}}{e^{x}-\sqrt{3}}\right|+C$

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Video Transcript

Okay, so this question wants us to evaluate this integral. So to do this, let's turn our expression into something that looks like our table so we can see how to easily do this by doing a little bit of pre processing on the denominator Eat of the two X is just eat of the ex quantity squared. So it seems like a natural substitution to make is for you to equal either the X because that means that d u is e to the x d X, which we already seeing are integral. So that means they're integral becomes the integral of well, eat of the ex d x. That's just do you over three minus you squared. And you can use partial fractions on this if you'd like. Or you could just look it up in an integral table and see this formula so I only have to do is find our value of a and we can plug in. So it says that our constant terms equals a squared. So three is equal to what squared well grew. Three. So a equals three. So this means our anti derivative becomes one over two times route three times the natural log of you, plus Route three over you minus or three plus c. And now we're pretty much done. We just need to plug back in our value of you, which was e to the X I'm in plus C, of course. So that's our final answer.

University of Michigan - Ann Arbor
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