We are given two integrals, and we aim to express the second one in terms of the first one. Let's break it down step by step.

1. First Integral Definition

We are provided with the integral:

$\alpha = \int_{\mathbb{R}^d} e^{-\frac{1}{2} \|x\|^2} \, dx$

This represents a Gaussian integral in $d$-dimensional space. The result of this integral is known and can be expressed as:

$\alpha = (2\pi)^{d/2}$

2. The Integral to Evaluate

We are tasked with evaluating the following integral:

$I = \int_{\mathbb{R}^d} e^{-\frac{1}{2}(x+b)^\top A (x+b)} \, dx$

where $A \in \mathbb{R}^{d \times d}$ is a positive-definite matrix, and $b \in \mathbb{R}^d$.

3. Change of Variables

To simplify the integral, we can perform a change of variables. Let:

$y = x + b$

Thus, $dy = dx$, and the integral becomes:

$I = \int_{\mathbb{R}^d} e^{-\frac{1}{2} y^\top A y} \, dy$

This is now in the form of a Gaussian integral but with the matrix $A$.

4. Known Result for Gaussian Integrals with Matrices

The general result for Gaussian integrals involving a positive-definite matrix $A$ is:

$\int_{\mathbb{R}^d} e^{-\frac{1}{2} x^\top A x} \, dx = \frac{(2\pi)^{d/2}}{\sqrt{\det(A)}}$

Thus, applying this result to our transformed integral:

$I = \frac{(2\pi)^{d/2}}{\sqrt{\det(A)}}$

5. Expressing in Terms of $\alpha$

We already know that $\alpha = (2\pi)^{d/2}$, so we can express the result for $I$ in terms of $\alpha$ as follows:

$I = \frac{\alpha}{\sqrt{\det(A)}}$

Final Answer

The value of the integral is:

$\boxed{\frac{\alpha}{\sqrt{\det(A)}}}$

Would you like more details on any part of the solution, or do you have further questions?

Here are 5 related questions to expand this topic:

1. How is the Gaussian integral derived in $d$-dimensions?
2. What are the properties of positive-definite matrices that are relevant to integrals?
3. How does the determinant of a matrix affect Gaussian integrals?
4. Can this method be extended to integrals with non-linear exponents?
5. What is the significance of the change of variables in multidimensional integrals?

Tip: In Gaussian integrals, a positive-definite matrix ensures convergence, allowing the use of known formulas for evaluation.