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Problem 42 Medium 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 x^2 \sqrt{1 - x^2}\ dx $

Answer

$\frac{x}{8}\left(2 x^{2}-1\right) \sqrt{1-x^{2}}+\frac{1}{8} \arcsin x+C$

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Video Transcript

Okay, So this question wants us to compute the following anti derivative using a computer algebra system. Compare that answer to the tabular form and show that they are indeed equivalent. So plugging into a computer algebra system, I was given this answer. But when plugging into a table, you're giving this answer. So let's try expanding the computer's answer and show that that's equal to the table. So all we gotta do is distribute this 1/8 teach term to expand everything out. And I'm just rearranging cause I looks more aesthetically pleasing. If you put the square root on the end and then, as you can see, this is really close. But there's just a factor of X in front of one of the terms, so we can just pull the X out and the other form has the term swapped over. So we'll do that. So all we have to do is pull out and acts from the first center parentheses. So we get to X squared minus one, keeping our square root intact plus 1/8 sign in verse of X plus C and comparing this one last time with our table. We see that the two answers are exactly the same. So our anti derivatives are, after all, equivalent

University of Michigan - Ann Arbor
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