When dealing with multi-dimensional integrations such as:
$$\int f\left(\sum_i x_i\right) \exp\left(\sum_i x_i^2 \right) d\vec x$$
where $f$ is a scalar function, I often find it helpful to make a change of variables using the basis:
$$u_1=(+1,+1,+1,...,+1) / \sqrt n$$ $$u_2=(+1,-1, 0,..., 0) / \sqrt 2$$ $$u_3=(+1, 0,-1,..., 0) / \sqrt 2$$ $$...$$ $$u_n=(+1, 0, 0,...,-1) / \sqrt 2$$
where $n$ is the dimension. After making a change of variables $\vec x\rightarrow \vec y$ using this basis, the above integral becomes:
$$\int f(\sqrt n y_1) \exp\left(\sum_i y_i^2\right) d\vec y$$
which can be better, because now the integrand factorizes over the $y_i$. Note that here the variable $y_1=\sum x_i\sqrt n$ is the one that does the trick, while the other basis vectors just have to be orthogonal to $u_1$ so that the Jacobian stays at 1.
I do this so often that it gets tiresome having to spell out the new basis. Is there name one can use for this kind of change of variables?