How do I integrate an exponential with a higher power?

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Homework Help Overview

The discussion revolves around integrating an exponential function multiplied by a polynomial term, specifically the integral of the form \(\int x^3 e^{-\alpha x^2} dx\). Participants are exploring techniques for handling this type of integral, which combines polynomial and exponential components.

Discussion Character

  • Exploratory, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • The original poster attempts to recall integration techniques and expresses uncertainty about the complexity of the problem. They mention trying substitution and express frustration with their lack of practice. Other participants suggest substitution as a viable approach and mention integration by parts as a method to simplify the integral.

Discussion Status

Participants are actively discussing potential methods for integration, with some suggesting substitution and integration by parts. There is acknowledgment of the original poster's initial thoughts and attempts, but no consensus has been reached regarding the best approach.

Contextual Notes

The original poster indicates a lack of recent practice with integration techniques, which may be influencing their confidence in solving the problem. There is also a mention of frustration with integration by parts, suggesting it may be a challenging aspect for them.

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Homework Statement


I'm being dead thick, but I can't remember how to integrate an exponential function.
[tex]\int x^3e^{-\alpha x^2}dx[/tex]

Homework Equations


The Attempt at a Solution


I reckon that this shouldn't be too complex, but I've totally forgotten how to go about this question. The [tex]x^3[/tex] term means that it can't be integrated like I would normally do (i.e. [tex]\int ke^x = ke^{\frac{x}{k}}[/tex]) can it? I've also tried substituting [tex]u=x^2[/tex] into it and following that through, but it doesn't seem to get anywhere.

Is this really simple, and my recent lack of practice with the techniques is just failing me, or there something more complicated about it?

Please assist, this is doing my head in!
 
Last edited:
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Substituting u = x^2 is a good first move.

You will get a simpler integral involving e^{-au} which you can solve by integration by parts.
 
[tex]\int x^3e^{-\alpha x^2}dx= \int x^2 e^{-\alpha x^2} (xdx)[/tex]

Now, as others have said, let u= x2.
 
Oh so I was going right.

I hate integration by parts.
 

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