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Problem 40 Easy Difficulty

Use a computer algebra system to evaluate the integral. Compare the answer with the result of using tables. If the answers are not the same, show that they are equivalent.

$ \displaystyle \int \frac{dx}{e^x (3e^x + 2)} $

Answer

$\frac{-e^{-x}}{2}+\frac{3}{4} \ln \left(2 e^{-x}+3\right)$

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Video Transcript

Okay, So this question wants us to compute this anti derivative using technology and comparing that to the table answer and showed that their equivalent So plugging into an integral calculator, I was given this answer from a computer algebra system. But when I plugged it into a table, I was given this answer. So as you can see these air pretty much the same answer. But let's just rearrange the CS answer. So we get negative e to the negative X over, too. Is there first term and then plus 3/4 times the natural log of to E to the minus. X plus three plus C. So this was a very straightforward calculation because their computer algebra system answer gave it in pretty much exact same form, as we see in the table right here. So since the answers match, we can say that the two anti derivatives were given are indeed equivalent

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