How do you use comparison test to determine is the integral is convergent or divergent given #int x / (8x^2 + 2x^2 - 1) dx# ?

Answer 1
First of all a think there is a mistake: maybe #8x^2# would be #8x^3#,

so the function is:

#x/(8x^3+2x^2-1)~1/(8x^2)# (#~ =#asymptotic)
that is a convergent function because it's a generalized armonic series, #1/x^alpha#, with #alpha>1#.
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Answer 2

To determine if the integral ∫(x / (8x^2 + 2x^2 - 1)) dx is convergent or divergent using the comparison test, follow these steps:

  1. Check if the given function x / (8x^2 + 2x^2 - 1) is non-negative and continuous on the interval of integration.

  2. Find a function g(x) such that 0 ≤ x / (8x^2 + 2x^2 - 1) ≤ g(x) for all x in the interval of integration, and g(x) is easier to integrate or determine the convergence of its integral.

  3. If the integral of g(x) converges, and if g(x) is greater than or equal to the given function for all x in the interval of integration, then the integral of the given function also converges. Conversely, if the integral of g(x) diverges, and if g(x) is less than or equal to the given function for all x in the interval of integration, then the integral of the given function also diverges.

  4. Evaluate the integral ∫g(x) dx and determine its convergence.

  5. Make a conclusion based on the comparison between the integral of the given function and the integral of g(x).

Apply the comparison test with a suitable function g(x) to determine the convergence or divergence of the given integral.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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