To find the integral of the function \(\frac{2}{9 + x^2}\), we can use a standard integration formula involving inverse trigonometric functions. 

### Integral Formula
The integral we need is:

\[
\int \frac{2}{9 + x^2} \, dx
\]

This is of the form \(\int \frac{a}{a^2 + x^2} \, dx\), which has a standard result:

\[
\int \frac{1}{a^2 + x^2} \, dx = \frac{1}{a} \arctan\left(\frac{x}{a}\right) + C
\]

### Applying the Formula
In our case, \(a^2 = 9\), so \(a = 3\). The integral can be rewritten as:

\[
2 \int \frac{1}{9 + x^2} \, dx = 2 \cdot \frac{1}{3} \arctan\left(\frac{x}{3}\right) + C
\]

Simplifying gives:

\[
\frac{2}{3} \arctan\left(\frac{x}{3}\right) + C
\]

### Final Answer
Thus, the integral of \(\frac{2}{9 + x^2}\) is:

\[
\int \frac{2}{9 + x^2} \, dx = \frac{2}{3} \arctan\left(\frac{x}{3}\right) + C
\]

Would you like to see more details, or do you have any questions?

### Related Questions:
1. How do you integrate \(\frac{1}{x^2 + a^2}\) in general?
2. What are the common integral forms involving inverse trigonometric functions?
3. Can you find the derivative of \(\arctan(x)\)?
4. How do we evaluate definite integrals involving \(\arctan(x)\)?
5. What is the importance of using trigonometric substitutions in integration?

**Tip:** Remember that recognizing patterns and forms is key to applying standard integration formulas correctly.

Find the integral of 2/(9 + x^2) using the standard formula for integrals involving inverse trigonometric functions. This calculus problem covers key concepts like integration techniques and the use of the arctan function, suitable for students in Grades 11-12 and early college.

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