How well can Mathematica do multivariate intergrating

Questions
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1

Dear Community,

At the moment I’m using Maple, however i’m not satisfied by the way it does multivariate integration and I’m wondering if I should switch to Mathematica.

Could someone tell me how Mathematica would deal with this simple example problem:
A typical example (from gaussian beam optics) is to find the normalization constant N for
E(x\_1)=N exp\[-ik x^T Q x ]
using the conservation of energy
1/N=\int \int E(x\_1) E(x\_1)^\* dx\_1dy\_1
where \* is complex conjugation,Q \el R^{4x4} and has the property that Q=Q^T and x=\[x\_1 y\_1 x\_2 y\_2 ].

Thanks!
Kind regards,
Carlas

2

Correction: E(x\_1)=N \int \int E\_0(x\_2) exp\[-ik x^T Q x ]dx\_2dy\_2

3

Could you translate the notation into something that does not assume as much background?

4

Multivariate integration by itself is no big deal. Examples are scattered through
Integrate: Compute definite, indefinite, region and other integrals—Wolfram Documentation

In[6]:= Integrate[ Sin*Sqrt, x, y]

Out[6]=
    2 3/2

  • ( - ) y Cos
        3

In[7]:= Integrate[ Sin[x*Sqrt], x, y]

Out[7]=
2 Sin[x Sqrt]
-( ---------------- )
            x