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$$\int{\exp((x_2-x_1)^2+k_1x_1+k_2x_2)dx_1dx_2}$$

I can perform the integration of the integral above easily by changing the variable \begin{align} u&=x_2+x_1\\ v&=x_2-x_1 \end{align} Of course first computing the Jacobian, and integrating over $u$ and $v$

I am wondering how you perform the change of variable for 4-dimensional integral like below:

$$\int{\exp\left(\sum_{i=1}^{4}((x_{i}-x_{i-1})^2+k_ix_i)\right)dx_1dx_2dx_3dx_4}$$

Is it something like: \begin{align} x_2-x_1&=u \\ x_2+x_1&=v \\ x_4-x_3&=p \\ x_4+x_3&=q \end{align} Should this be enough? I think one only needs 4 new variables right? Because I was thinking the integral would be easier if I could do something like: \begin{align} x_2-x_1&=u \\ x_3-x_2&=v \\ x_4-x_3&=p \\ x_1-x_4&=q \end{align} Or in general, how do you perform change of variable in multi-dimensional case. Should you generally perform it by doing:

\begin{align} u&=a_1x_1+a_2x_2+a_3x_3+a_4x_4 \\ v&=b_1x_1+b_2x_2+b_3x_3+b_4x_4 \\ p&=c_1x_1+c_2x_2+c_3x_3+c_4x_4 \\ q&=d_1x_1+d_2x_2+d_3x_3+d_4x_4 \end{align}

And what kind of change of variable does one need to perform to perform the integration of such integration easily?

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    I notice that in your first attempt to substitute four variables, you have $x_3-x_2=\frac12(q-p-v-u),$ which seems worse than you started with, but in your second attempt the four variables are linearly dependent: $u+v+p+q=0,$ so the substitution may not work (depending on the $k_i$). Things are easier if you have only $n-1$ differences of the form $x_i-x_j$ for $n$ variables (as you do in the first integral), rather than $n$ differences (as you have in the second integral). – 2017-02-12
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    If $A$ is a real, symmetric and positive definite $n\times n$ matrix, $$\int_{\mathbb{R}^n}\exp\left(-x^T A x\right)\,d\mu = \frac{\pi^{n/2}}{\sqrt{\det A}}.$$ – 2017-02-12
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    Anyway, it would be extremely useful to know the integration bounds. – 2017-02-12

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