https://file.aiquickdraw.com/mathbot/file/1726199819317hvgkobyt.jpg

The integral in the image is:

\[
\int \frac{1}{\sqrt{x^2 + 9}} \, dx
\]

To solve this, we can use a standard integral formula. This resembles the derivative of the inverse hyperbolic sine function, specifically:

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

In our case, \(a = 3\), so the solution becomes:

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

Alternatively, this can be expressed as:

\[
\ln \left( x + \sqrt{x^2 + 9} \right) + C
\]

Thus, the final answer is:

\[
\ln \left( x + \sqrt{x^2 + 9} \right) + C
\]

Would you like a more detailed explanation or have any questions?

Here are 5 related questions for further exploration:
1. How do you compute the integral of \(\frac{1}{\sqrt{x^2 - a^2}}\,dx\)?
2. What are inverse hyperbolic functions and their derivatives?
3. How would you approach integrals involving \(\frac{1}{x^2 + a^2}\)?
4. Can you use trigonometric substitution to solve this integral? If so, how?
5. How do you find the derivative of the natural logarithm of a function?

**Tip**: Recognizing patterns in integrals is key to solving them efficiently, especially when inverse functions or logarithmic forms are involved.

Compute the integral of 1 / sqrt(x^2 + 9) dx.

Learn how to compute the integral of 1 / sqrt(x^2 + 9). This solution uses inverse hyperbolic functions and logarithmic forms, offering a clear step-by-step explanation. Ideal for students in Calculus I/II.

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