How do you determine if the improper integral converges or diverges #int [(x^3)( e^(x^4) )] dx# from negative infinity to infinity?
see below
For starters, it's actually a fairly tame function that you are looking to integrate
it is the product of 2 continuous functions, and that is a very important factor
we could explore the derivative to check for weird behaviour, but we may as well look at the [indefinite] integral as it is surprisingly easy:
there is a pattern:
Now applying the limits of integration:
not surprising once you see this
graph{x^3 e^(x^4) [2.002, 2.003, 1.001, 1.002]}
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To determine if the improper integral ( \int_{\infty}^{\infty} x^3 e^{x^4} , dx ) converges or diverges, we need to evaluate the integral separately for each half of the interval and then consider their combined behavior.

For the lower limit, ( \infty ), we substitute ( t = x^4 ), and as ( x ) approaches ( \infty ), ( t ) approaches ( +\infty ). Thus, the integral becomes ( \int_{+\infty}^0 \frac{e^t}{4t^{3/4}} , dt ).

For the upper limit, ( +\infty ), we substitute ( u = x^4 ), and as ( x ) approaches ( +\infty ), ( u ) approaches ( +\infty ). Thus, the integral becomes ( \int_0^{+\infty} \frac{e^{u}}{4u^{3/4}} , du ).
Both integrals can be evaluated using techniques such as comparison tests or integration by parts. If both integrals converge, the original improper integral converges; if at least one of them diverges, the original integral diverges.
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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