I am guessing that you wanted to change all
\(40)
to
(
and
\(41)
to
)
and
e
to
E
and
Integrate[...,{x3,x2,3000},{x2,x1,3000},{x1,0,3000}]
to
Integrate[...,{x1,0,3000},{x2,x1,3000},{x3,x2,3000}]
If all that is correct then your input becomes
Integrate[(E^((-1/9*(-3 + x1)^2 - (-8 + x2)^2/16 - (-12 + x3)^2/36)/2)*
(7.5*x1 + 6*(-x1 + x2) + 4.5*(-x2 + x3)))/(144*Sqrt[2]*Pi^(3/2)),
{x1,0,3000},{x2,x1,3000},{x3,x2,3000}]
Then this
Table[x1=RandomReal[{0,3000}];x2=RandomReal[{x1,3000}];x3=RandomReal[{x2,3000}];
(E^((-1/9*(-3 + x1)^2 - (-8 + x2)^2/16 - (-12 + x3)^2/36)/2)*
(7.5*x1 + 6*(-x1 + x2) + 4.5*(-x2 + x3)))/(144*Sqrt[2]*Pi^(3/2)),
{1000}]
returns this
{0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0....
seeming to show that your integrand is very small, almost zero, almost everywhere.
It appears that only for small values of x1,x2,x3, less than 50, is your integrand slightly larger than zero.That appears to be confusing the integration algorithm, where it looks at values of your integrand hundreds or thousands of times and finds the value is almost exactly zero every time.
If all of my changes and guesses are correct then this
NIntegrate[(E^((-1/9*(-3 + x1)^2 - (-8 + x2)^2/16 - (-12 + x3)^2/36)/2)*
(7.5*x1 + 6*(-x1 + x2) + 4.5*(-x2 + x3)))/(144*Sqrt[2]*Pi^(3/2)),
{x1,0,40},{x2,x1,40},{x3,x2,40}]
estimates your full integral is approximately 38.2249 but I do not know how much error is associated with that.
Please check every detail of this and try to make certain that you have found and corrected all my mistakes.