# How do you determine if the improper integral converges or diverges #int (x^3 + x)/((x^4 + 2x^2 + 2)^(1/2))dx# from 1 to infinity?

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To determine if the improper integral converges or diverges, we first find its limits of integration.

lim as t approaches infinity of ∫(x^3 + x)/((x^4 + 2x^2 + 2)^(1/2))dx from 1 to t.

We then use a suitable convergence test. In this case, we can use the limit comparison test with the function 1/x^2.

Taking the limit as t approaches infinity of the ratio of the two functions, we have:

lim as t approaches infinity of ((x^3 + x)/((x^4 + 2x^2 + 2)^(1/2))) / (1/x^2)

After simplification, we obtain:

lim as t approaches infinity of x^5/(x^4 + 2x^2 + 2)^(1/2)

Since the degree of the numerator is greater than the degree of the denominator, this limit will diverge. Therefore, by the limit comparison test, the original integral also diverges.

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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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