https://file.aiquickdraw.com/mathbot/file/17262769132372ydfeik8.jpg

The integral shown in the image is:

\[
\int_{16}^{25} \frac{\sqrt{x}}{x - 9} \, dx
\]

This is an improper integral due to the potential singularity at \(x = 9\) in the denominator. We can approach this integral using substitution and partial fraction methods, but we should first analyze whether there are any singularities in the domain of integration. Specifically, we'll check for issues where \(x - 9 = 0\), which occurs at \(x = 9\).

Let me go ahead and solve this step by step.

### Steps for solving:

1. **Substitution**: One possible substitution is to let \(u = \sqrt{x}\), which simplifies the square root and the integrand.

2. **Handling the singularity**: Since \(x = 9\) is within the integration bounds, we need to determine whether the integral converges.

I'll work through these steps now.The evaluated integral is:

\[
\int_{16}^{25} \frac{\sqrt{x}}{x - 9} \, dx = 2 + 3 \log(7) - 3 \log(8) + 3 \log(2)
\]

This can be simplified further:

\[
2 + 3 \log \left(\frac{7 \times 2}{8}\right)
= 2 + 3 \log \left(\frac{14}{8}\right)
= 2 + 3 \log \left(\frac{7}{4}\right)
\]

This represents the exact solution of the integral. Let me know if you'd like further steps or a numerical approximation.

Would you like a breakdown of the steps, or do you have any other questions?

### Related Questions:
1. What methods can be used to solve integrals with singularities?
2. How do you determine if an improper integral converges?
3. Can partial fraction decomposition be applied to this problem?
4. How does the substitution method help in simplifying integrals involving square roots?
5. What are some alternative ways to approximate difficult integrals numerically?

### Tip:
Always check for singularities within the integration limits, as they can cause improper integrals to diverge or require special handling.

Evaluate the integral \(\int_{16}^{25} \frac{\sqrt{x}}{x - 9} \, dx\).

This page provides a detailed solution to the improper integral \(\int_{16}^{25} \frac{\sqrt{x}}{x - 9} \, dx\), using substitution and singularity analysis. The integral is evaluated step-by-step, with explanations on handling potential singularities at x = 9.

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