The integral you have shared can be interpreted as:

$\int_{\mathbb{R}^{n-1} \setminus B(0,1)} \frac{1}{(\lambda^2 + y^2)^{\frac{n}{2}}} \, dy$

Key elements:

1. The region of integration is over $\mathbb{R}^{n-1} \setminus B(0,1)$, meaning it is taken over $(n-1)$-dimensional Euclidean space $\mathbb{R}^{n-1}$ minus the ball of radius 1 centered at the origin $B(0,1)$.
2. The integrand is $\frac{1}{(\lambda^2 + y^2)^{\frac{n}{2}}}$, where $\lambda$ is a constant and $y$ is a variable in $\mathbb{R}^{n-1}$.
3. $dy$ refers to the differential in the $(n-1)$-dimensional space.

This is a typical form for integrals in higher-dimensional spaces, and appears often in the context of Green's functions, potential theory, or heat kernels.

To evaluate this integral, we would typically employ spherical coordinates in $\mathbb{R}^{n-1}$. The volume element in spherical coordinates in $\mathbb{R}^{n-1}$ can simplify the integral, especially since the integrand depends only on the radial component $y^2$. Here's a rough sketch of the steps:

Steps:

1. Switch to Spherical Coordinates: Let $y \in \mathbb{R}^{n-1}$, and set $y = r$, where $r = |y|$ is the radial distance.

   The volume element in $\mathbb{R}^{n-1}$ in spherical coordinates becomes $dy = r^{n-2} \, dr \, d\Omega$, where $d\Omega$ is the differential solid angle over the unit sphere $S^{n-2}$.

2. Simplify the Integrand: The integrand becomes $\frac{1}{(\lambda^2 + r^2)^{\frac{n}{2}}}$.

3. Reduce the Integral: The angular part can be factored out, leaving an integral over the radial coordinate $r$ from 1 to $\infty$ (due to the exclusion of the unit ball $B(0,1)$).

4. Compute the Radial Integral: The remaining integral over $r$ will involve standard techniques for integrals of this form, often involving substitutions and special functions like Beta or Gamma functions.

Would you like me to proceed with the full evaluation of this integral, or would you like clarification on any specific step so far?

Related Questions:

1. What is the physical or mathematical context behind this integral? Is it related to a specific problem?
2. Would you like a more detailed explanation of the switch to spherical coordinates in $\mathbb{R}^{n-1}$?
3. Would you like an introduction to using special functions (like the Gamma function) to evaluate these integrals?
4. How does the exclusion of the ball $B(0,1)$ affect the convergence or solution?
5. Do you need this integral in terms of specific parameters, like $\lambda$, or in terms of a general solution?

Tip: For integrals in higher dimensions, switching to spherical or polar coordinates can often significantly simplify the problem, especially when dealing with radial symmetries.