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The result of the integral 

\[
\int_0^9 \frac{dt}{\sqrt{81 + t^2}}
\]

is \( \log(1 + \sqrt{2}) \).

Would you like a detailed explanation of how this solution is derived?

Here are 5 related questions for further exploration:
1. What is the general method to evaluate integrals involving square roots?
2. How do logarithms appear in the solutions of integrals like this?
3. Can this integral be solved using a trigonometric substitution?
4. What are some real-world applications of integrals like this one?
5. How would the result change if the limits of integration were different?

**Tip:** When evaluating integrals with square roots, try substitutions like \( t = a \tan(\theta) \), as they simplify the expression.

Evaluate the integral from 0 to 9 of dt over the square root of (81 + t^2).

This problem involves evaluating the definite integral from 0 to 9 of dt over sqrt(81 + t^2). The solution is derived using trigonometric substitution, resulting in a logarithmic expression. This problem is suitable for advanced calculus students.

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