Certainly! The expression $\sin(\pi y)$ can be simplified before integration. Recognizing that $\pi y$ is just a linear transformation of $y$, we can use a substitution to make the integral more straightforward.

Suppose we want to integrate $\sin(\pi y)$ with respect to $y$. Here's a step-by-step approach:

1. Substitution: Let $u = \pi y$. Then, $du = \pi \, dy$ or $dy = \frac{du}{\pi}$.

2. Rewrite the Integral: $\int \sin(\pi y) \, dy = \int \sin(u) \cdot \frac{du}{\pi}$

3. Simplify the Integral: $\int \sin(\pi y) \, dy = \frac{1}{\pi} \int \sin(u) \, du$

4. Integrate: The integral of $\sin(u)$ with respect to $u$ is $-\cos(u)$: $\frac{1}{\pi} \int \sin(u) \, du = \frac{1}{\pi} (-\cos(u)) = -\frac{1}{\pi} \cos(u)$

5. Substitute Back: Recall $u = \pi y$: $-\frac{1}{\pi} \cos(u) = -\frac{1}{\pi} \cos(\pi y)$

Therefore, the integral of $\sin(\pi y)$ with respect to $y$ is: $\int \sin(\pi y) \, dy = -\frac{1}{\pi} \cos(\pi y) + C$ where $C$ is the constant of integration.

Would you like further details or have any questions about this process?

Here are some related questions:

1. How do you integrate other trigonometric functions?
2. What are common techniques for integrating products of trigonometric functions?
3. Can you explain the method of integration by parts?
4. How do you handle integrals involving trigonometric identities?
5. What are some real-world applications of integrating trigonometric functions?

Tip: When dealing with trigonometric integrals, look for opportunities to simplify using trigonometric identities or substitutions to make the integral more manageable.