101. Many methods needed Show that the integral from ∫(from 0 to ∞)(sqrt(x) * ln x) / (1 + x)^2 dx equals pi, following these steps
d. Evaluate the remaining integral using the change of variables z = sqrt(x)

Verified step by step guidance

1. Start with the integral after previous simplifications, which should be of the form \(\int_0^{\infty} \frac{\sqrt{x} \ln x}{(1+x)^2} \, dx\).
2. Apply the substitution \(z = \sqrt{x}\), which implies \(x = z^2\). Then, compute the differential \(dx\) in terms of \(dz\): \(dx = 2z \, dz\).
3. Rewrite the integral in terms of \(z\) by substituting \(x = z^2\), \(\sqrt{x} = z\), \(\ln x = \ln(z^2) = 2 \ln z\), and \(dx = 2z \, dz\). The integral becomes \(\int_0^{\infty} \frac{z \cdot 2 \ln z}{(1 + z^2)^2} \cdot 2z \, dz\).
4. Simplify the integrand by combining terms: \(z \cdot 2 \ln z \cdot 2z = 4 z^2 \ln z\), so the integral is \(\int_0^{\infty} \frac{4 z^2 \ln z}{(1 + z^2)^2} \, dz\).
5. Now, the integral is expressed in terms of \(z\) and can be evaluated using appropriate methods such as integration by parts or recognizing it as a standard integral form.

Key Concepts

Change of Variables (Substitution) in Integration

This technique involves replacing the original variable with a new variable to simplify the integral. By setting z = sqrt(x), we rewrite the integral in terms of z, which often makes the integral easier to evaluate. It requires adjusting the differential dx accordingly and changing the limits of integration.

Improper Integrals over Infinite Intervals

Integrals with limits extending to infinity are called improper integrals. Evaluating them requires understanding limits and convergence. Here, the integral from 0 to ∞ must be handled carefully, ensuring the integral converges and applying appropriate techniques to evaluate it.

Logarithmic Functions within Integrals

Integrals involving logarithmic terms, such as ln(x), often require special attention due to their behavior near zero and infinity. Understanding properties of logarithms and how they interact with other functions in the integrand is essential for simplifying and evaluating the integral.