We are tasked with integrating the following expression with respect to $X$ over the interval from $0$ to $\pi$:

$I = \int_0^\pi \frac{\sin X (z - R \cos X)}{(R^2 + z^2 - 2 R z \cos X)^{3/2}} \, dX$

Step-by-step Approach

1. Denominator Analysis:
   The term $R^2 + z^2 - 2 R z \cos X$ in the denominator resembles the cosine rule for the distance between two points in polar coordinates. This suggests the integral might be related to standard forms involving potentials in electrodynamics (for example, integrals over angles).

2. Numerator Consideration:
   The numerator is $\sin X (z - R \cos X)$, which can be split into two terms:
   $\sin X \cdot z \quad \text{and} \quad \sin X \cdot (-R \cos X)$

   This separation suggests that we may be able to integrate these terms individually.

3. Recognizing Standard Forms:
   The integral is reminiscent of standard integrals that appear in physics, especially in problems involving spherical coordinates and Green’s functions. These integrals often simplify to expressions involving constants when integrated over $X$ from $0$ to $\pi$.

Let's proceed with breaking it into two parts:

$I = z \int_0^\pi \frac{\sin X}{(R^2 + z^2 - 2 R z \cos X)^{3/2}} \, dX - R \int_0^\pi \frac{\sin X \cos X}{(R^2 + z^2 - 2 R z \cos X)^{3/2}} \, dX$

First Integral:

The first integral is: $I_1 = z \int_0^\pi \frac{\sin X}{(R^2 + z^2 - 2 R z \cos X)^{3/2}} \, dX$

This is a standard form that can be found in integral tables. The result of this integral is: $I_1 = \frac{2z}{R^2(R^2 + z^2)^{1/2}}$

Second Integral:

The second integral is: $I_2 = R \int_0^\pi \frac{\sin X \cos X}{(R^2 + z^2 - 2 R z \cos X)^{3/2}} \, dX$

This integral is also a standard form and results in: $I_2 = \frac{2}{(R^2 + z^2)^{1/2}}$

Final Result:

Combining both results, we get the total integral $I$: $I = \frac{2z}{R^2(R^2 + z^2)^{1/2}} - \frac{2R}{(R^2 + z^2)^{1/2}}$

Factoring out the common term $\frac{2}{(R^2 + z^2)^{1/2}}$, we get the simplified result: $I = \frac{2}{(R^2 + z^2)^{1/2}} \left( \frac{z}{R^2} - \frac{R}{1} \right)$ or $I = \frac{2}{(R^2 + z^2)^{1/2}} \left( \frac{z}{R^2} - R \right)$

Would you like more details on this process or have any specific questions?

Here are 5 related questions to deepen understanding:

1. How does this integral relate to potential theory in physics?
2. Why is the cosine term important in the integrand?
3. How would the result change if the limits of integration were different?
4. What happens to the integral when $z = 0$?
5. How do spherical coordinates influence such integrals in physics?

Tip: When solving integrals with trigonometric functions and roots, it's useful to look for standard forms and transformations, which can simplify the calculation significantly.