https://file.aiquickdraw.com/mathbot/file/1726623075562zn4zbkvm.jpg

The image contains an improper integral:

\[
\int_{3}^{\infty} \frac{5}{x^2} \, dx
\]

### Solution:

1. First, rewrite the integral:

\[
\int_{3}^{\infty} \frac{5}{x^2} \, dx = 5 \int_{3}^{\infty} x^{-2} \, dx
\]

2. Find the antiderivative of \( x^{-2} \):

\[
\int x^{-2} \, dx = \frac{-1}{x}
\]

3. Apply the limits of integration:

\[
5 \left[ \frac{-1}{x} \right]_{3}^{\infty} = 5 \left( 0 - \frac{-1}{3} \right) = 5 \times \frac{1}{3}
\]

4. Simplifying:

\[
5 \times \frac{1}{3} = \frac{5}{3}
\]

Thus, the value of the improper integral is \( \frac{5}{3} \).

Would you like more details or have any further questions?

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Here are 5 related questions to further your understanding:
1. What happens if we change the lower limit of integration?
2. How do we evaluate improper integrals with different types of singularities?
3. What does it mean for an improper integral to converge?
4. How would this integral change if the exponent on \(x\) was different?
5. Can improper integrals always be solved analytically?

**Tip:** When solving improper integrals, always check for convergence first before proceeding to evaluate.

Evaluate the improper integral \( \int_{3}^{\infty} \frac{5}{x^2} \, dx \).

This problem involves solving the improper integral \( \int_{3}^{\infty} \frac{5}{x^2} \, dx \) using techniques of integration and evaluating limits. Explore step-by-step how to rewrite the integral, find the antiderivative, and apply limits of integration to determine convergence. Suitable for undergraduate calculus students.

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