Help me please with some integrals

[thenerdystudent]

I have attached a file with the problem I have to solve. Thank you in advance for anyone who can help

 

[Deleted member 4993]

I have attached a file with the problem I have to solve. Thank you in advance for anyone who can help

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[stapel]

I have attached a file with the problem I have to solve. Thank you in advance for anyone who can help

The image is too small for all of the text to be clear. I think the following is what is contained in the image:

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Solve two integrals.

The following two integrals are extended to the three-dimensional space \(\displaystyle \, \mathbb{R}:\)

. . . . .\(\displaystyle \displaystyle N\, =\, \int_{\mathbb{R}^2}\, \dfrac{R\, (x\, \sin(\theta)\, -\, y\,\cos(\theta)}{(R\, +\, x\, \sin(\theta)\, -\, y\, \cos(\theta))^4}\, dV\)

. . . . .\(\displaystyle \displaystyle M\, =\, \int_{\mathbb{R}^2}\, \dfrac{R^2}{(R\, +\, x\, \sin(\theta)\, -\, y\, \cos(\theta))^4}\, dV\)

...where \(\displaystyle \, dV\, =\, dx\, dy\, dz\,\) is the volume element and:

. . . . .\(\displaystyle \displaystyle R\, =\, \sqrt{\strut \left(x\, -\, \cos(\theta)\right)^2\, +\, \left(y\, -\, \sin(\theta)\right)^2\, +\, z^2\,}\)

...is the distance from any point \(\displaystyle \, (x,\, y,\, z)\, \in\, \mathbb{R}^3\,\) to a point \(\displaystyle \,\left(\cos(\theta),\, \sin(\theta),\, 0\right)\,\) on the unit circle on the XOY plane and centered at the origin, which itself depends on (x, y, z) by the expression \(\displaystyle \, \theta\, =\, -R.\)

Properties

• It is suspected that both integrals give the same value.
• The integrals are singular only along the points on the XOY plane which satisfy

. . . . .\(\displaystyle x(\phi)\, =\, \cos(\phi)\, -\, \phi\, \sin(\phi),\, y(\phi)\, =\, \sin(\phi)\, +\, \phi\, \cos(\phi),\)

...and depict the espiral of step \(\displaystyle \,2\pi.\)

Problem

Determine, by any method, the finite value of both integrals N and M, either analytically or numerically, if the singularities can be circumvented.

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When you reply with your thoughts and efforts so far, kindly please also include corrections or confirmation. Thank you!