# Re: Calculus problem

```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           /
\------------------------------------------------------------------------/

```

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