Final answer:
The divergence of the integral of g(x) does not provide enough information to determine the convergence of the integral of f(x).the correct answer to the question 'If ∫ g(x)dx is divergent, then ∫ f(x)dx is?' is D) Insufficient information.
Step-by-step explanation:
When considering the convergence of an indefinite integral of the form ∫ f(x)dx, where a is a constant and f(x) is a function, the divergence or convergence of a separate function g(x) does not necessarily imply the same for f(x). Each function must be evaluated on its own merits, as they can behave quite differently. For example, ∫ 1/x2dx is convergent, while ∫ 1/xdx is divergent, though both integrals seem similar at a glance.
As a result, without additional information about the behavior of f(x) itself, one cannot determine the convergence of ∫ f(x)dx solely based on the divergence of ∫ g(x)dx. Therefore, the correct answer to the question 'If ∫ g(x)dx is divergent, then ∫ f(x)dx is?' is D) Insufficient information.