PSA: @Stephen_Wolfram just released a notebook version of WolframAlpha. I can't tell you how many times I've used WolframAlpha as an undergrad learning more advanced math, or as a researcher when dealing with some very very ugly integral. https://blog.stephenwolfram.com/2019/09/the-ease-of-wolframalpha-the-power-of-mathematica-introducing-wolframalpha-notebook-edition/ …
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Yeah, doesn't MIT have a Mathematica site license? Why use WolframAlpha when you can use Mathematica?
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Great Q. I use Mathematica when I want to perform some very complicated symbolic-type operation to test some hypothesis I have. Or, in one case, to assist in a proof (see http://math.mit.edu/~urschel/instabilities.pdf …). One of the biggest benefits of WolframAlpha for me is interface and ease of use.
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