Re: Integral of exp(sin(x)) dx


"Chris Rodgers" <rodgers@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx> wrote in message
news:eo35b5$rhv$1@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Hi,

I have a problem which involves finding integrals of the form

/ x=X
|
| exp(sin(x)) dx
|
/ x=0

or

/ x=X
|
| exp(a+b*x+sin(c*x)) dx
|
/ x=0

where a,b,c are real numbers.

Does anyone have any suggestions on how such an integral might be tackled
symbollically? I have already tried Maple and Mathematica on these with no
success.

Alternatively, is there a compelling reason why such integrals CANNOT be
found (as opposed to my just not being smart enough to do them)?


Sure. There are many functions that do not have an "elementary"
anti-derivative. See http://en.wikipedia.org/wiki/Differential_Galois_theory
..

Usually in this case one "artificially" defines the anti-derivative to be
the anti-derivative of that function. i.e., its really for symbolic reasons
and to express over functions in terms of the anti-derivative in a
simplified way(so you don't have to write the integral out every time).

In a sense though even functions like sin(x) are not elementary and have no
elementary derivatives. For example, compute int(sin(x),x=0..1) exactly?
well its obviously cos(1) but what is cos(1)? Do you know exactly what it
is? (maybe, there are many formulas to find many "special" values of the
trig functions but AFAIK there are an infinite number of arguments that do
not produce exact results from finite means(you can compute them with
infinite series but in that sense they are not elementary). Nevertheless we
still think of integrals of these trig functions as elementary because they
are so common and do have special properties that makes them easy to work
with.

(I'm not sure if the above is entirely correct but I think it gets the jist
of it. I think it all boils down to being able to compute the values at had
using a finite number of additions(when all operations are converted to
addition). If you can't then its not elementary. (I could be wrong here but
this is how I tend tot hink of it))

Jon


.

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