The problem asks to compute the integral with an accuracy of $\epsilon = 10^{-3}$. The integral is:

$\int_0^{3/4} \frac{\ln(1 + x^2)}{x} \, dx$

Let's solve this step-by-step.

Step 1: Analyze the nature of the integrand

The integrand is $\frac{\ln(1 + x^2)}{x}$, which becomes problematic at $x = 0$ because the denominator becomes zero. We need to analyze the behavior near $x = 0$.

At $x = 0$: \frac{\ln(1 + x^2)}{x} \approx \frac{x^2}{x} = x \quad \text{(for small values of x)}. Thus, near $x = 0$, the integrand behaves like $x$, which is integrable.

Step 2: Numerical Integration

Since the problem asks for an approximate solution with precision $\epsilon = 10^{-3}$, a numerical method such as Simpson's rule, trapezoidal rule, or Gaussian quadrature would be appropriate.

I will now compute the value of this integral numerically.The approximate value of the integral is $0.249$ with an error smaller than $10^{-15}$, which is well within the required precision of $\epsilon = 10^{-3}$.

Would you like more details about the steps involved, or do you have any questions?

Here are 5 related questions for further exploration:

1. How does the integrand behave as $x \to 0$ and $x \to \infty$?
2. What is Simpson's rule, and how does it compare to Gaussian quadrature?
3. Can the integral be evaluated analytically, or is numerical integration always required for such cases?
4. How do we handle singularities at the boundary when computing integrals numerically?
5. What other numerical integration methods are suitable for improper integrals?

Tip: When performing numerical integration, ensure that the chosen method handles potential singularities effectively for accurate results.