2. When integrating by parts the aim is to obtain a simpler integral than the one we started off with. 3. When integrating by parts the aim is to obtain a recursive relation between integrals.

Both statements you provided about integration by parts are correct, but they emphasize different aspects of the technique. 1. **Simplifying the Integral**: The primary goal of integration by parts is often to transform a complex integral into a simpler one. By choosing appropriate functions \( u \) and \( dv \), you can sometimes reduce the degree of the polynomial or the complexity of the function being integrated, making it easier to evaluate. 2. **Recursive Relation**: Integration by parts can also lead to a recursive relationship between integrals. This is particularly useful when dealing with integrals of the same form that can be expressed in terms of one another. By applying integration by parts multiple times, you can derive a formula that relates the integral of a function to itself, which can be solved iteratively. In summary, integration by parts serves both to simplify integrals and to establish recursive relationships, depending on the context and the functions involved.