Re: Calculus problem
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Re: Calculus problem
> jhanson <jhanson@CSCI.CSUSB.EDU> wrote:
> jhanson <jhanson@CSCI.CSUSB.EDU> wrote:
> : Someone please integrate this by hand: sin(3x^2)
> I got this formula out of the CRC "Standard Mathematical Tables and
> Formulae", 29th Edition. (If you don't have this book, I highly suggest it
> for reference.) Please excuse the poor rendition in text-only mode.
> - Inf
> / 1
> | sin ax^n dx = -------- * GAMMA(1/n) * sin (PI/2n)
> / na^(1/n)
> - 0
> (This is the definite integral from 0 to Infinity, for n > 1)
> Where GAMMA(1) = 1
> GAMMA(1/2) = sqrt(PI)
> GAMMA(n+1) = n*GAMMA(n) for n > 0
> So, we get that the integral of sin(3x^2) as evaluated from 0 to infinity
> equals:
> 1/(2*3^(1/2)) * GAMMA(1/2) * sin(PI/2*2)
> = sqrt(PI)/(2*sqrt(3)) * sin(PI/4)
> I couldn't find reference to a non-definite (ie. open) integral. Perhaps
> it doesn't exist in closed form, but I think you could evaluate the
> integral by using parts (even if it is messy).
> /------------------------------------------------------------------------\
> / Terry Fleury - tfleury@uiuc.edu - http://www.students.uiuc.edu/~tfleury/ \
> \ "Give me the credit (with interest, please)." - Wir /
> \------------------------------------------------------------------------/
The function sin(3x^2) does not have an elementary antiderivative.
This means there is no expression formed by combination or composition of
elementary functions which has sin(3x^2) as its derivative. To really
appreciate this fact, one needs to understand the use of "elementary" here:
it means algebraic functions (formed by addition, subtraction, multiplication
or division -- including powers -- and the inverses of such functions), the
trigonometric or circular functions (any function that can be derived by
algebraic processes from sine) and their inverses, general exponential
functions (to an arbitrary positive base) and their inverses (logartihmic
functions).
That's nothing unusual -- in fact when you start randomly forming functions by
combining or composing elementary functions, _most_ of the results you arrive
at do not have elementary antiderivatives. (One would have to define what
"most" means in a sentence like this in order to defend it, of course!).
The converse is not the case -- the derivative of an elementary function is an
elementary function. This (rather surprising) fact plus the fact that
techniques of calculating derviatives tend to be studied first in traditional
and you are much further ahead than you used to be.
calculus courses tends to hide the fact that anti-derivation is a much more
general operation than derivation. Attention has tended to be focused on the
rather small class of elementary functions for which it _is_ possible, by
means of substitution etc, to wrestle out an expression for the antiderivative
in elementary functions. Of course, this class of functions includes many of
those that are of direct interest in simple applications, and learning to apply
these simple functions to, say, physical or financial problems whose solution
involves finding derivatives or antiderivatives seems an appropriate
introduction to the actual use of calculus.
But the other, more stubborn functions whose antiderivatives are not elementary
also come up frequently in important applications. Engineers and applied
mathematicians very early start working with Bessel functions, the gamma
function, elliptic functions, hypergeometric functions, Hankel functions, etc
etc. The process is very simple -- take an elementary function -- say sin(x^2)
-- which has no elementary antiderivative, and give a _name_ to its
antiderivative. You have a new (non-elementary function). Then, by using
processes of substitution, you can identify a whole passel of elementary
functions whose antiderivatives can be expressed in terms of your new function
(possibly in combination with other already-established classes of functions),
and you are much further ahead than you used to be.
You really have to understand how functions work in order to be able to do this
-- as well as be in control of the algebraic processes involved. I am not
currently responsible for teaching calculus, but if I were I would be pushing
my students toward this level of understanding and competence rather than
focusing on the traditional content of an integral calculus course -- methods
of somehow grinding out messy antiderivatives for the restricted class of
functions in which the result is elementary (but not necessarily simple).
This is a skill of rapidly decreasing practical importance, overtaken by
technology just as many of the old skilled handtrades have been. I am afraid,
however, that this viewpoint is still far from universal adoption, and we still
see a whole generation being raised in the belief that there _must_ somehow be
an elementary antiderivative for such a simple function as sin(3x^2), if only
we were smart enough to be able to find it....
RWW Taylor
National Technical Institute for the Deaf
Rochester Institute of Technology
Rochester NY 14623
>>>> The plural of mongoose begins with p. <<<<
P.S. If you look at the _graph_ of sin(3x^2) you can easily see that it does
not make sense to try to integrate it "from zero to infinity" as is
suggested in the quote. These formulas have to be taken with a grain of
salt (and common sense).