i have this integral , i will use s for the integration sign: 
(2/pi)S e^(-x/2) * cos(2nx)dx (from 0 to pi) , 
the answer should be 0.504*(2/(1+16n^2))
However when i use the ti 89 the answer that i get is different,
anybody know how to do it?

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Clarification of Question by sergiobenavides-ga on 01 Dec 2005 14:45 PST

hfshaw-ga:

You seems to know what you are doing. Thats true going to the
indefinite integrals tables and pluggin values, you will get that
answer. However just to clarify my question , HOW DO I SOLVE  a
indefinite integral USING THE TI-89?

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Clarification of Question by sergiobenavides-ga on 06 Dec 2005 04:04 PST

It seems like nobody knows. I finally figure it out. The ti89 show you
that answer because N is not defined, so the only extra step needed is
to define n as an integer int(z)-->n and it will work..... int is in
catalog and ---> is STO-->

If you have written the integral properly, the answer you think
"should" be correct, isn't.

From a table of indefinite integrals:

Integral of {exp(ax)*cos(bx) dx} = 

exp(ax)*[a*cos(bx) + b*sin(bx)]/(a^2 + b^2) + constant

in your case, a = -1/2, and b = 2*n.  Plug in these values, and then
use the limits of integration to turn the indefinite integral result
into a definite integral result.    That results in:

(2/pi)*[exp(-pi/2)*(-0.5*cos(2*pi*n) + 2*n*sin(2*pi*n)) -
exp(0)*(-0.5*cos(0) + 2*n*sin(0)]/[1/4 + 4*n^2]

Now, exp(0) = 1, cos(0) = 1, and sin(0) = 0.  Furthermore, for any
integral value of n (you didn't specify that n must be integral in
your question, but I assume this is the case), cos(n*2*pi) = 1, and
sin(n*2*pi) = 0.  The result then simplifies to:

(2/pi)*[exp(-pi/2) * (-1/2) + 1/2]/[1/4 + 4*N^2]

From a table of indefinite integrals:

Integral of {exp(ax)*cos(bx) dx} = 

exp(ax)*[a*cos(bx) + b*sin(bx)]/(a^2 + b^2) + constant

in your case, a = -1/2, and b = 2*n.  Plug in these values, and then
use the limits of integration to turn the indefinite integral result
into a definite integral result.    That results in:

(2/pi)*[exp(-pi/2)*(-0.5*cos(2*pi*n) + 2*n*sin(2*pi*n)) -
exp(0)*(-0.5*cos(0) + 2*n*sin(0)]/[1/4 + 4*n^2]

Now, exp(0) = 1, cos(0) = 1, and sin(0) = 0.  Furthermore, for any
integral value of n (you didn't specify that n must be integral in
your question, but I assume this is the case), cos(n*2*pi) = 1, and
sin(n*2*pi) = 0.  The result then simplifies to:

(2/pi)*[exp(-pi/2) * (-1/2) + 1/2]/[1/4 + 4*N^2]

Multiplying through by 4/4 and simplifying results in:

(2/pi)* [-2*exp(-pi/2) + 2]/[1+16*n^2].

Evaluating the exponential and doing a little more arithemetic results
in your answer:

0.50428*(2/(1+16*n^2))

Ignore that first comment.  I hit "post" by mistake.

Before I give the derivation a shot, let me confirm with you what I
got on my calc (I'm using a Voyage200, OS 3.10).  It behaves just like
an 89 with the exception of a larger screen and a qwerty keyboard. 
There have been some changes in the algebra handling since earlier
versions of the OS, so we may not be getting the same result.


according to Buttons (that's his name, ya know):
                                              -pi                    -pi
       pi                                     ---                    ---
            -x                                 2                      2
 2     SS  ---                 -4*cos(2n*pi)*e      16*n*sin(2*n*pi)*e         4
--- *  S    2  * cos(2nx) dx = ----------------- +
-------------------- + --------------------
pi    SS  e                     (16*n^2+1)*pi       (16*n^2+1)*pi     
    (16*n^2+1)*pi

(i really hope this won't wordwrap, expect a followup if it does)

I have yet to find the calculator wrong in its integration (more often
than not it takes into account factors integration tables ignore).  I
wouldn't give it much praise for simplifying the answers well,
though...

I hope this will help any researchers that may not have immediate
access to an advanced graphing calculator.

Okay...fixed up for your viewing pleasure:



       pi                                     
            -x                                 
 2     SS  ---                 
--- *  S    2  * cos(2nx) dx = [continued below?]
pi    SS  e                     

       0

              -pi                    -pi
              ---                    ---
               2                      2
-4*cos(2n*pi)*e      16*n*sin(2*n*pi)*e           4
----------------- +  -------------------- + --------------
   (16*n^2+1)*pi       (16*n^2+1)*pi        (16*n^2+1)*pi