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Problem 39 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{x^2 + 4}\ dx $

Answer

$$\frac{1}{4} x\left(x^{2}+2\right) \sqrt{x^{2}+4}-2 \ln (\sqrt{x^{2}+4}+x)+C$$

Discussion

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Video Transcript

Okay, so this question wants us to evaluate this anti derivative using a computer algebra system and show us that that's the same thing that we get if we use the table. So I plugged into a computer algebra system and got this answer. But when you plug into a table, you get this form. So let's show they're the same, but converting the computer form into the tabular for So let's distribute this 1/4 to each term. So we get square root X squared, plus four over four times Ex cued plus two X minus eight over for Ellen of X plus Square root X squared plus four plus C. Then we can factor out X from the first term. So we get X over four times X squared plus two X minus. Sorry. Need our square roots still minus two. Ellen of X was squared, X squared, plus floor plus c. And then this looks really close to our table for him. We just need to pull out a factor of 1/2 from the first term. So we get X over eight times. Well, if we pulled out a factor on half if to multiply, both things and the parentheses by two, and we leave the second term alone, and this should be our final answer. So let's compare this. We get X over a times two X squared plus four that are square term. Yep, Then minus to Ellen of X plus Squared X squared plus four, which we do indeed get, making sure to expand just for clarity and the answer's air equivalent.

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