Solve Integral A(x)*x^2*cos(nx) w/ Mathematica

[randomvar]

Hi guys,
I need a hint on how to solve this integral using mathematica

I = Integral ( A(x)*x^2*cos(nx) )dx , (0 to infinity)

A is a function of x and infact i have the values of A for different values of x which was obtained experimentally.

I need to find the value of I(the integral) for many values of n say from (n=1 to 250)

Any thoughts would be greatly helpful. Let me know if the question is not clear

Thanks a lot,

 

[Negatron]

Integrate[A (x)*x^2*Cos[n*x], {x, 0, Infinity}] does not converge so no definite integral... if I understood that correctly.

edit: Integrate[a[x]*x^2*Cos[n*x], {x, 0, Infinity}]. Or is this what you want? a is a function of x in this one not a variable. Here too you will not get a definite integral.

 

[Hepth]

Only if A is a negative exponential will it converge :

[tex]
A =\sum _{i=1}^{\infty } A_i e^{-c_i x^i}
[/tex]

 

[Simon_Tyler]

Since you have experimental values of A, construct an interpolating function then do the numerical integral. Eg

pts=RandomReal[{0,100},50]//Sort;
data=Table[{x,(100+RandomReal[{-.01,.01}])Exp[-.01 x^2]},{x,pts}];
A=Interpolation[data]
Plot[{A[x],100 Exp[-.01 x^2]},{x,0,100},PlotRange->All,PlotStyle->{Automatic,Dashed}]

But note that the interpolation function is only good within your data range.

Plot[A[x],{x,99,1001},PlotRange->All]
A[100000000000]

But if A[x] really does drop off exponentially (as Hepth points out that it must), then you're probably ok to truncate the integral - eg

int=Table[NIntegrate[A[x] x^2 Cos[n x],{x,0,100}],{n,1,100}]
ListPlot[int[[1;;20]],Joined->True,PlotRange->All]

If you have some more information and can guess what the functional form of A(x) is, then it might be better to use the function Fit[data,funs,vars] to the data instead of interpolate.

 

[blue_raver22]

To solve this integral in Mathematica, you can use the built-in function "Integrate" and specify the limits of integration as well as the function A(x). For example, if A(x) is a polynomial function, you can use the following code:

Integrate[A[x]*x^2*Cos[n*x], {x, 0, Infinity}]

If A(x) is a more complex function, you may need to use numerical methods to approximate the integral. In that case, you can use the "NIntegrate" function and specify the number of points to use for the approximation. For example:

NIntegrate[A[x]*x^2*Cos[n*x], {x, 0, Infinity}, Method -> "GaussKronrodRule", MaxRecursion -> 20]

You can then use a loop to iterate through different values of n and obtain the corresponding values of the integral. I hope this helps!