To solve the integral of \( \frac{1}{x^2} \sin \left( \frac{1}{x} \right) \, dx \), we can use substitution.

### Steps:

Let:
\[
u = \frac{1}{x}
\]
Then:
\[
du = -\frac{1}{x^2} dx
\]
Thus, the integral becomes:

\[
\int \frac{1}{x^2} \sin \left( \frac{1}{x} \right) dx = - \int \sin(u) \, du
\]

Now, we integrate \( \sin(u) \):

\[
- \int \sin(u) \, du = \cos(u) + C
\]

Finally, substitute \( u = \frac{1}{x} \) back into the result:

\[
\cos \left( \frac{1}{x} \right) + C
\]

So, the integral of \( \frac{1}{x^2} \sin \left( \frac{1}{x} \right) \, dx \) is:

\[
\cos \left( \frac{1}{x} \right) + C
\]

Would you like further details on substitution or have any other questions?

Here are 5 related questions to expand on this topic:
1. How does substitution work in integrals?
2. What is the integral of \( \frac{1}{x^2} \cos\left( \frac{1}{x} \right) \)?
3. How do you handle improper integrals involving trigonometric functions?
4. What happens if you differentiate the result of this integral?
5. How would you compute this integral with limits of integration?

**Tip:** When stuck on a complex integral, substitution is often a helpful technique to simplify the expression.

Learn how to solve the integral of (1/x^2)sin(1/x) using substitution. This step-by-step guide explains how to simplify the integral with substitution and trigonometric integration, perfect for undergraduate calculus students.

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