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If ∫[a, [infinity]] g(x)dx is divergent, then ∫[a, [infinity]] f(x)dx is:

A) Divergent
B) Convergent
C) Indeterminate
D) Insufficient information

User Boggio
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1 Answer

6 votes

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.

User ShapCyber
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