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.