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Given:
\[
I(x) = \int_{1}^{\infty} \frac{\ln(t)}{t + x} dt
\]

Next steps appear to involve differentiation with respect to \( x \) (likely applying Leibniz's rule):

1. The first step involves differentiating the integral with respect to \( x \):
\[
I'(x) = -\int_{1}^{\infty} \frac{\ln(t)}{(t + x)^2} dt
\]

2. The next step seems to evaluate the derivative of \( I'(x) \) again:
\[
I''(x) = \int_{1}^{\infty} \frac{2 \ln(t)}{(t + x)^3} dt
\]

Would you like me to explain these steps in more detail, or help with solving or understanding any part of this problem?

Here are some related questions you might consider:

1. How does Leibniz's rule apply to differentiating under the integral sign?
2. What are the conditions for differentiating an integral with respect to a parameter?
3. How can we interpret the behavior of the function \( I(x) \) as \( x \) approaches certain limits?
4. What techniques can be used to solve integrals that involve logarithmic functions?
5. How can one verify the convergence of the given improper integral?

**Tip:** When differentiating under the integral sign, always check if the conditions of Leibniz's rule are satisfied, especially the uniform convergence of the integral with respect to the parameter being differentiated.

Understand how to differentiate an integral involving logarithms step-by-step using Leibniz's rule. This problem involves integral calculus and explores differentiation techniques with detailed examples and theoretical insights.

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