On 13/12/2010 08:36 AM, scott wrote:
>> Having just said that, as best as I can tell, taking the derivative
>> /usually/ makes something simpler, and taking the integral therefore
>> /usually/ makes something more complicated.
>
> Derivatives are easy if you have systematic brain that can follow
> logical methods. You learn a few simple rules and then you are able to
> pretty much differentiate any function, no matter how complex (it just
> takes up more sheets of paper).
>
> Integration on the other hand requires you to have a different skill,
> one to figure out what function might differentiate to give you back
> your original function.

Indeed, there appears to be a *method* for figuring out a derivative. 
The only method for finding an integral seems to be trying expressions 
at random until one yields the derivative you want. There doesn't seem 
to be any more direct method.

> There are a few common procedures and rules of
> thumb, but faced with an unfamiliar form often requires a bit or trial
> and error or luck... Luckily at school we had a table of common
> functions and their integral, and I suppose the questions were
> specifically designed to make use of these. In the real world you seem
> to get stuck very quickly though and just rely on a computer :-)

Like I say, it appears that "in the real world", the integral usually 
doesn't have a closed form at all.

For example, Wolfram Alpha claims that there is no closed-form integral 
for tanh(x^2). And... well, check this out:

http://www.wolframalpha.com/input/?i=Integrate[x+Sin[x]+Cos[x^2]%2C+x]

Post a reply to this message