Final answer:
Integration by parts is used to solve the integral of z(ln z)² dz, by setting one part as u and the other as dv then finding the corresponding du and v to substitute back into the formula.
Step-by-step explanation:
To integrate z((ln z)² dz, we use integration by parts, which is a technique applicable to the product of two functions. The formula for integration by parts is \(∫ u dv = uv - ∫ v du\). In this context, let's set \(u = (ln z)²\) and \(dv = z dz\). Then differentiate and integrate to find \(du = 2 (ln z) \frac{dz}{z}\) and \(v = \frac{z²}{2}\). After substituting back into the integration by parts formula, we just need to simplify and integrate any remaining terms.
This technique is a part of calculus and is critical in solving integrals that cannot be evaluated using basic integration rules. It's often used in applications involving physics or engineering where expressions contain products of functions.