We are tasked with finding the values of $p$ for which the given integral converges:

$\int_0^{1/2} \frac{dx}{5x (\ln(x))^p}$

Step 1: Understanding the behavior near the bounds

• At the upper bound $x = 1/2$, the logarithmic term $\ln(x)$ does not approach zero or infinity. Hence, there is no issue with convergence here.
• The critical issue arises at the lower bound $x = 0$, where $\ln(x) \to -\infty$, so we must focus on the behavior near $x = 0$.

Step 2: Analyze the singularity near $x = 0$

To study the behavior near $x = 0$, substitute $u = \ln(x)$, where $du = \frac{dx}{x}$. Thus, the integral becomes:

$\int_{-\infty}^{\ln(1/2)} \frac{du}{5u^p}$

We now analyze the integral for the convergence of:

$\int_{-\infty}^{\ln(1/2)} \frac{du}{u^p}$

Step 3: Convergence conditions

For the integral $\int \frac{du}{u^p}$ to converge as $u \to -\infty$:

• If $p > 1$, the integral diverges because the decay is not strong enough to make the area finite.
• If $p = 1$, the integral behaves like $\ln(u)$ near $u \to -\infty$, which also diverges.
• If $p < 1$, the integral converges, because the decay is strong enough to make the area finite.

Conclusion:

The integral converges when $p < 1$.

Would you like further clarification or details on any part? Here are some related questions:

1. How does logarithmic behavior impact integrals at infinity?
2. What are the general conditions for the convergence of improper integrals?
3. How does changing the limits of integration affect convergence in integrals with singularities?
4. Can we generalize this problem to other logarithmic powers?
5. What are other types of singularities that affect integral convergence?

Tip: Always analyze the behavior of an integrand near the problematic bounds, such as $0$ or $\infty$, when determining convergence.