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To solve the given integral:

\[
\int \frac{dx}{36 + x^2}
\]

we'll use a standard formula for integrating a function of the form:

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

where \( a \) is a constant.

Here, we have \( a^2 = 36 \), so \( a = 6 \).

Applying the formula:

\[
\int \frac{dx}{36 + x^2} = \frac{1}{6} \tan^{-1}\left(\frac{x}{6}\right) + C.
\]

The correct answer is:

\[
\boxed{\text{C: } \frac{1}{6} \tan^{-1}\left(\frac{x}{6}\right) + C}
\]

Would you like more details or have any questions?

### Related Questions:
1. How do you integrate functions of the form \( \int \frac{dx}{a^2 + x^2} \)?
2. What is the geometric interpretation of the arctangent function?
3. Can you solve an example where the integral is \( \int \frac{dx}{25 + x^2} \)?
4. What happens if \( a \) is a function of \( x \) instead of a constant in \( \int \frac{dx}{a^2 + x^2} \)?
5. How does the result change if we integrate \( \int \frac{x \, dx}{36 + x^2} \)?

**Tip:** Remember that integrals involving \( a^2 + x^2 \) often lead to inverse trigonometric functions like \( \tan^{-1} \).

Evaluate the integral: ∫ dx / (36 + x^2).

Learn how to solve the integral ∫ dx / (36 + x^2) using the inverse trigonometric formula. This step-by-step solution covers key calculus concepts, including integration techniques and inverse trigonometric functions. Suitable for students in Grades 11-12.

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