Integrate:

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

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?

## Comments

Sort by:

TopNewestSame 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

Log in to reply

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

Log in to reply

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

Log in to reply

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

Log in to reply

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

Log in to reply

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

Log in to reply

– Michael Mendrin · 2 years, 11 months ago

I'm waiting for the time when they could put one of those chips in my head.Log in to reply