# How do you determine if the improper integral converges or diverges #int 5x^(2)e^(-x^(3))# from 1 to infinity?

Hence, the integral converges,

and the value of the integral is

As this is an improper integral (we have an infinite upper bound), we must examine the behaviour of the integral at the upper limit:

And so:

So for our improper integral we have:

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To determine if the improper integral ( \int_{1}^{\infty} 5x^2e^{-x^3} , dx ) converges or diverges:

- First, check for convergence at ( x = 1 ). Evaluate ( 5x^2e^{-x^3} ) at ( x = 1 ) to see if it's finite.
- Next, investigate the behavior of the integrand as ( x ) approaches infinity. Analyze whether ( 5x^2e^{-x^3} ) approaches zero as ( x ) goes to infinity, as this is a necessary condition for convergence.
- If the integrand approaches zero as ( x ) approaches infinity, apply the Limit Comparison Test or the Comparison Test with a known convergent or divergent integral to determine convergence or divergence of the original integral.
- If the integral passes both convergence tests, then it converges. If it fails one or both tests, it diverges.

By following these steps, you can determine if the improper integral converges or 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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