It seems you've uploaded an image. I'll extract and analyze the mathematical content from it. Let me examine the image to see what it contains.The image contains a mathematical expression involving integrals. Here’s a transcription of the visible part of the image:

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.