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\(\int { \frac{\sin^3{x}\cos^3{x}}{\sin^3{x}+\cos^3{x}+\tan^3{x}} } dx\)

No, please don't do that. Allow your friend Wolfram to do it. But how does IT do it?

Solving this integral by hand would take beyond eternity. But with a little computer programming, BAM!... you've got the answer in 2 seconds.

This can be viewed as an example of what can be accomplished if one does not approach the problem directly. Instead, you go invent computers, and have them do it for you. In the future, cybernetic engineering just might allow people to install Wolfram Alpha chips into their brains and solve those dam integrals... Wouldn't that be AWESOME? Just imagine solving this integral in 2 seconds!! But... showing the steps later on to your teacher could be a bit of an issue...

...which brings up the next point: What work does Wolfram Alpha do?

Note by John Muradeli
2 years, 11 months ago

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Same kind of work what hand calculators do, which used to take days or even years to do in the past centuries. Now we can more efficiently design things because we have stuff like computers and software. Knowing how to integrate is one skill, but mathematics is a whole universe of skills, most of it still not implemented by computers or software.

For example, I used Mathematica just now to work out that integral, and it gave a slightly different result than the one delivered by Wolfram Alpha (in spite of the fact both software are Wolfram!). You need to have a good understanding of mathematics to even understand how they are related, and no computer or software can ever do that for you. Michael Mendrin · 2 years, 11 months ago

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@Michael Mendrin Oh and yeah you're right like with that "UN-Integratable" solution by Hassan I was starting to write a rant about how his answer is wrong because it didn't match Wolfram's, even after I differentiated, but then after some more playing around I found that his answer is also right and just just the same version of a different thing. I quickly erased the comment, glad he didn't see it (but he may know now that I've said it).

but still, they do give out a general answer, right? How do you program a machine to do something you can't do?

SKYNET CONFIRMED!!! John Muradeli · 2 years, 11 months ago

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@John Muradeli Mathematica integrated that "un-integrable" integral with no problem, but what I found to be an interesting read was the different ways the answer can be expressed. So, still, sometimes people can deliver cleaner or better results than math software.

One who is familiar with mathematics knows how often seemingly different things turn out to be exactly the same. To me, one of the most fascinating things about theoretical physics is how often this happens---where seemingly disparate theories with different math in them turn out to be mathematically equivalent. I wish more would be said or publicized about this in physics. Michael Mendrin · 2 years, 11 months ago

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@Michael Mendrin Hm, yes, but W|A does output an answer for the indefinite integral, which means it has to get it from somewhere other than brute numerical calculations. If it only did definite integrals and approximations, like you with Mathematica and 100,000 points trapezoidal rule, I'd understand that since it's just calculating power. But W|A, once again, gives the general answer, meaning it doesn't do just approximations and number crunching, but it actually finds the indefinite integral.

So how does it do it? And how do you program a machine to do such a thing? It probably can't be done with a python... can it? All I know is rect[1,10] and stuff like that from 3 hour lecturing in Khan Academy, and apparently it takes about 10,000 lines of code to make a simulation of a 2D flower growing from a flowerpot. LOL... John Muradeli · 2 years, 11 months ago

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@John Muradeli Mathematica did give the exact answer to the indefinite integral, just in a slightly different form. How does it do it? Through "symbolic processing", which is a generalization of what hand calculators do with number crushing. If you peruse through Mathematica commands, there's a ton of commands that deal with symbolic manipulation, which is covered first before you get into mathematics. Much like Wikipedia, many mathematicians have voluntarily contributed to the power of Wolfram software by developing techniques that are implementable by Wolfram software, so that when presented with something to solve or computer, such as a indefinite integral, it will try a number of tactics based on those techniques---and only when none of them work will the software return the original statement with no results. It's a massive software, and its library continues to grow yearly.

Stephen Wolfram is convinced that he will usher in "a new kind of science". Well, perhaps that's a bit grandiose, but for sure computers and software are changing the way we think about science and mathematics, and our role in it as "mere humans". Michael Mendrin · 2 years, 11 months ago

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@Michael Mendrin Aah so that's why my TI-89 keeps returning stuff back to me? It's so annoying I can't even describe it >.<

So, you're saying, there are special techniques that only high computational power entities can employ? Like, we can do IBP, but never a hand limit of trapezoid rule as # of trapezoids approaches infinity, or something like that? I see.

Alright, thank you Michael. And watch out for those Wolfram|Chips\(^{TM}\) in the future! John Muradeli · 2 years, 11 months ago

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@John Muradeli I'm waiting for the time when they could put one of those chips in my head. Michael Mendrin · 2 years, 11 months ago

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