The given expression is \( \int \sin(\pi y) \cdot \frac{dy}{y} \).

This is a standard integral that can be solved using substitution and known integral forms. The solution involves recognizing the logarithmic form for the integration of \( \sin(\pi y) / y \).

### Solution:

1. The integral is:
   \[
   \int \frac{\sin(\pi y)}{y} \, dy
   \]

2. This is a standard result known as the **Sine Integral**, often denoted as \( \text{Si}(x) \) for \( x = \pi y \).

Thus, the result is:

\[
\text{Si}(\pi y) + C
\]

where \( \text{Si}(x) = \int_0^x \frac{\sin t}{t} \, dt \), and \( C \) is the constant of integration.

### Final Answer:
\[
\int \frac{\sin(\pi y)}{y} \, dy = \text{Si}(\pi y) + C
\]

Would you like to explore this result further or any specific part of it?

### Related Questions:
1. What is the sine integral function and where is it used?
2. Can you provide an example of evaluating \( \text{Si}(x) \) at specific values of \( y \)?
3. How would you approach solving \( \int \cos(\pi y)/y \, dy \)?
4. What are the applications of the sine integral in physics or engineering?
5. How does the sine integral relate to Fourier transforms?

### Tip:
Sine and cosine integrals often appear in problems involving waveforms, signal processing, and diffraction patterns.

Explore the detailed solution to the integral of sin(πy)/y, which involves the sine integral function, denoted as Si(πy). Learn the mathematical concepts behind special functions, integral calculus, and their applications in physics and engineering. Suitable for undergraduate math students.

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