To integrate \( y = 3 \cos\left(\frac{2}{x}\right) \), we'll use a substitution method. Let's follow the steps below:

1. **Substitute**: Let \( u = \frac{2}{x} \). Then \( du = -\frac{2}{x^2} dx \) or \( dx = -\frac{x^2}{2} du \).

2. **Rewrite**: Notice that \( x = \frac{2}{u} \), so \( dx = -\frac{4}{u^2} du \).

   Now, substitute these into the integral:
   \[
   \int 3 \cos\left(\frac{2}{x}\right) dx = \int 3 \cos(u) \left(-\frac{4}{u^2}\right) du
   \]
   Simplify:
   \[
   \int 3 \cos(u) \left(-\frac{4}{u^2}\right) du = -6 \int \frac{\cos(u)}{u^2} du
   \]

3. **Integrate**: We now need to integrate \( \int \frac{\cos(u)}{u^2} du \). This integral is not straightforward and typically requires special functions or integration techniques. It is known to involve the Cosine Integral function \( \text{Ci}(x) \), but for educational purposes, let's express it in terms of known integrals.

4. **Express**: The integral can be expressed as:
   \[
   \int \frac{\cos(u)}{u^2} du = \int \frac{\cos(u)}{u^2} du = \text{Ci}(u)
   \]
   Hence,
   \[
   -6 \int \frac{\cos(u)}{u^2} du = -6 \text{Ci}(u)
   \]

5. **Back-substitute**: Replace \( u \) with \( \frac{2}{x} \):
   \[
   -6 \text{Ci}\left(\frac{2}{x}\right)
   \]

Thus, the integral of \( y = 3 \cos\left(\frac{2}{x}\right) \) is:

\[
\int 3 \cos\left(\frac{2}{x}\right) dx = -6 \text{Ci}\left(\frac{2}{x}\right) + C
\]

where \( C \) is the constant of integration.

Do you want details or have any questions?

Here are 8 related questions to expand your understanding:

1. What is the Cosine Integral function, and how is it defined?
2. How do you integrate \( \sin\left(\frac{2}{x}\right) \) using substitution?
3. Can we use partial fractions in trigonometric integrals?
4. What is the integral of \( \cos\left(\frac{k}{x}\right) \) for any constant \( k \)?
5. How does substitution simplify complex integrals?
6. What are special functions in mathematics?
7. How do you verify the result of an integral using differentiation?
8. What are common techniques for integrating trigonometric functions?

**Tip:** When integrating complex expressions, substitutions can simplify the process by transforming the integral into a more manageable form.

Learn how to integrate y = 3*cos(2/x) step-by-step using the substitution method. This problem involves advanced calculus techniques and trigonometric functions. Suitable for advanced high school and college students.

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