# How do you use the integral test to determine whether #int lnx/(xsqrtx)# converges or diverges from #[3,oo)#?

The integral converges to

Let's use integration by parts. Let:

So the (presently unbounded) integral becomes:

As we've already determined:

Thus, the improper integral is:

The integral converges.

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To determine the convergence or divergence of the integral ∫(ln(x) / (x√x)) from [3, ∞), you can use the integral test. First, verify that the function f(x) = ln(x) / (x√x) is continuous, positive, and decreasing for x ≥ 3. Then, evaluate the improper integral ∫(ln(x) / (x√x)) from 3 to ∞. If the integral converges, the series also converges. If the integral diverges, the series 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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