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

In summary, the conversation discusses the integration of a complex function involving a cubic and square root term. The person was initially trying to use substitution to integrate it, but it did not work. They then mention using numerical methods like Simpson's rule or elliptic integrals, but it is concluded that the problem cannot be solved in closed form and can only be evaluated numerically. The problem was originally encountered when trying to find the surface of revolution of a cubic function.
  • #1
flash
68
0
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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  • #2
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.
 
  • #3
You can painfully do it by elliptic intehrals. But how did you arrive at this problem?
 
  • #4
I put it into maple, and the answer cannot be expressed with elementary functions (and the answer is about 19 lines!)
 
  • #5
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.
 
  • #6
How did you evaluate an indefinite integral numerically :(
 
  • #7
Haha it was a definite integral, just didn't bother with the notation :-)
 

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

What is a complex polynomial?

A complex polynomial is a mathematical expression made up of variables, coefficients, and constants, all of which can be complex numbers. It is in the form of zn + an-1zn-1 + ... + a1z + a0, where n is a positive integer, z is a complex variable, and a0, a1, ..., an-1 are complex numbers.

What does it mean to integrate a complex polynomial?

Integrating a complex polynomial means finding the antiderivative of the polynomial. It is the inverse operation of differentiation, and it involves finding a function that, when differentiated, gives the original polynomial.

How do you integrate a complex polynomial?

To integrate a complex polynomial, you can use the standard integration rules for polynomials, such as the power rule, constant multiple rule, and sum rule. You can also use the Cauchy-Riemann equations to find the antiderivative of a complex polynomial.

What are some applications of integrating complex polynomials?

Integrating complex polynomials is useful in many areas of mathematics and science, such as in solving differential equations, finding areas and volumes of complex shapes, and in signal processing and control systems.

Are there any special cases when integrating a complex polynomial?

Yes, there are special cases when integrating a complex polynomial, such as when the polynomial has singularities, poles, or branch points. In these cases, special techniques, such as the residue theorem or contour integration, may be used to find the antiderivative.

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