It seems like you're asking for the integral of \( \frac{6}{1 + x^2} \, dx \).

This is a standard integral involving the inverse tangent function. The formula for the integral of \( \frac{1}{1 + x^2} \, dx \) is:

\[
\int \frac{1}{1 + x^2} \, dx = \tan^{-1}(x) + C
\]

So, multiplying by 6:

\[
\int \frac{6}{1 + x^2} \, dx = 6 \tan^{-1}(x) + C
\]

### Final answer:
\[
6 \tan^{-1}(x) + C
\]

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

Here are 5 related questions:
1. What is the geometric interpretation of the inverse tangent function?
2. How would you solve integrals involving higher powers of \(x^2\)?
3. Can you extend this result to functions like \( \frac{1}{x^2 + a^2} \)?
4. What is the derivative of the arctangent function?
5. How can you apply this in physics problems, particularly in projectile motion?

**Tip:** Always check for trigonometric identities when you encounter expressions like \(1 + x^2\) in integrals.

Learn how to solve the integral ∫ 6/(1 + x^2) dx using the inverse tangent function. This problem involves inverse trigonometric integration and is suitable for undergraduate calculus students.

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