In Exercises 69–80, determine whether the improper integral co...

Question

In Exercises 69–80, determine whether the improper integral converges or diverges. If it converges, evaluate the integral.

∫₋∞⁰ x² e^(x³) dx

Accepted Answer

Hello there, today we are going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Determine whether the improper integral converges or diverges and find its value if it converges. The integral evaluated from zero to infinity of x 2 multiplied by e to the power of neg x 3 dx. OK. So it appears for this particular prompt we are given an improper integral and we're asked to determine if this improper integral will converge or diverge, and if it does converge, we're asked to find its value of convergence. So ultimately, simply put, we are determining whether or not this improper integral converges or diverges. So now that we know what we're ultimately trying to solve for, our first step that we need to take is we need to rewrite our improper integral as a limit. So we need to take the integral, evaluate from 0 to infinity of x 2 multiplied by e to the power of x 3. The x which will be equal to the limit as b approaches infinity of the integral evaluated from zero, the b of x to the power of 2 multiplied by e to the power of negative x to the power of 3. D X. So now, at this point, we need to be able to look at our limit, and we will note that with the integral that we are dealing with, we will need to recall and apply the integration by parts method. So with that in mind, let us state that you. Will be equal to x to the power of 3, which will mean therefore that dU by dx, so duu by dx will be equal to -3 multiplied by x to the 2, which will mean that x 2 dx will be equal to -1 divided by 3 duu. So now we can begin to evaluate our integral, so we could take the integral of x 2 multiplied by e x 3. The x, which will be equal to 1/3. Multiplied by the integral of e to the power of u multiplied by d u, which will be equal to 1/3, so 1/3 multiplied by e to the power of u, plus c where c denotes some constant variable, and this will be equal to 1/3 multiplied by e to the power of x to the power of 3. Plus C. Thus we can go ahead and write that the integral evaluated from 0 to b of x 2 multiplied by e to the power of x 3. The X will be equal to 1/3 multiplied by e to the x 3, evaluated from 0 to B. Will be equal to 1/3 multiplied by square brackets of e to the b 3 minus e to the power of 03. Which will be equal to 1/3 multiplied by square brackets of 1 divided by e to the power of b to the power of 3 minus 1, which will be equal to 1/3 minus 1 divided by 3 multiplied by e multiplied, so it's going to be 3 multiplied by e to the power of b to the power of 3. OK. So we now need to evaluate our limit. So we need to take the limit as b approaches infinity of parentheses of 1/3 minus 1 divided by 3 multiplied by e to the power of b to the power of 3. Which will be equal to square brackets of 1 divided by 3. -1 divided by 3 multiplied by e to the power of infinity to the power of 3. Which will be equal to 1/3 sense. Let's make a note up here in blue that sense 1 divided by 3 multiplied by e to the power of infinity to the power of 3 will approach 0. So therefore we could safely say that our final answer will be that it converges. So once again it will converge to 1/3, and that's it. We've officially solved for this problem. Hooray. So looking at the prom itself, we can safely say once again that our final answer is converges to 1/3 or 1 divided by 3. And that's it. That is our final answer. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

In Exercises 69–80, determine whether the improper integral converges or diverges. If it converges, evaluate the integral.∫₋∞⁰ x² e^(x³) dx

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In Exercises 69–80, determine whether the improper integral converges or diverges. If it converges, evaluate the integral.
∫₋∞⁰ x² e^(x³) dx