Final answer:
To choose between integration by substitution or integration by parts, consider the complexity of the integrand and the geometry of the path. Use substitution if the integrand simplifies with a change of variable, and integration by parts if the integral involves a product of functions where one part is easily differentiable and the other easily integrable.
Step-by-step explanation:
The subject of this question relates to determining whether integration by substitution or integration by parts is more appropriate for given integrals. When dealing with integrals involving vector quantities and geometric paths, we consider the nature of the integrand and the path over which the integral is taken. Often, these types of integrals can be simplified by expressing them in terms of a single variable, which can lead to a more straightforward integration process.
For example, reducing a line integral over a parabolic path to an integral in terms of x may simplify the integral if the corresponding function in x is simpler than its y-counterpart, which may involve complex operations like extracting square roots or dealing with fractional exponents.
If the integrand and the differential element (dx, dy, etc.) can be related through a simple substitution, integration by substitution is usually the more suitable method. On the other hand, if the integral involves a product of functions where one function is easily differentiable and the other is easily integrable, integration by parts might be the better choice. If neither of these situations applies, a different integration method may be required.