How Do You Integrate a Cubic Polynomial with a Complex Radicand?

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Homework Help Overview

The discussion revolves around the integration of a cubic polynomial multiplied by a complex radicand, specifically involving the integral of a cubic polynomial and a square root expression. The problem raises questions about the feasibility of finding a closed-form solution.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • The original poster attempts to use substitution based on the derivative of the radicand but finds it unworkable. Other participants suggest that closed-form integration may not be possible and mention numerical methods like Simpson's rule. There is also a mention of elliptic integrals as a potential approach, though it is noted to be complex.

Discussion Status

The discussion is ongoing, with participants exploring different methods and expressing skepticism about the possibility of a closed-form solution. Some guidance has been offered regarding numerical methods, and there is acknowledgment of the complexity involved in the problem.

Contextual Notes

Participants note that the problem arose in the context of evaluating the surface of revolution of the cubic function, and there is a clarification regarding the nature of the integral being definite rather than indefinite.

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Hi
I'm trying to integrate
[tex] \int (-40x^3 + 38.4x^2 - 13.288x + 1.98072)\sqrt{14400x^4 - 18432x^3 + 9087.36x^2 - 2041.0368x+ 177.570944}dx[/tex]
The way I thought I could do it was express the first part (the cubic) in terms of the derivative of the second and do it by substitution. Unfortunately it doesn't work. :frown:
Not totally sure that it can be done at all.

Thanks for any help!
 
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No, I don't think you are going to be able to integrate that in closed form. But there are simple numerical methods to consider like Simpson's rule.
 
You can painfully do it by elliptic intehrals. But how did you arrive at this problem?
 
I put it into maple, and the answer cannot be expressed with elementary functions (and the answer is about 19 lines!)
 
Ok thanks people. I found the answer numerically. I arrived at the problem when trying to evaluate the surface of revolution of the cubic function.
 
How did you evaluate an indefinite integral numerically :(
 
Haha it was a definite integral, just didn't bother with the notation :-)
 

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