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This integral poses a challenge.

Ordinarily integrating rational functions can be solved using the Hermite-Ostrogradski method. However, in the following integral, the coefficients $\beta_0, ..., \beta_4$ are not integers. (Hence, the Hermite-Ostrogradski method would not be appropriate).

Note: Expanding the integrand (trying to solve the integral using a partial fraction decomposition) is (because of the nature of the physical problem described by this integral) an inappropriate solution to this case.

$$\int \dfrac{1}{\beta_0 + \beta_1x + \beta_2x^2 + \beta_3x^3 +\beta_4x^4} dx$$

How can this rational function be evaluated?

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  • $\begingroup$ why partial fraction decomposition is inappropriate? $\endgroup$ – Szeto Jun 29 '18 at 23:18
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    $\begingroup$ I do not think that you could do it without partial fraction decomposition (at least in the genral case). $\endgroup$ – Claude Leibovici Jun 30 '18 at 4:03
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Note: Expanding the integrand (trying to solve the integral using a partial fraction decomposition) is (because of the nature of the physical problem described by this integral) an inappropriate solution to this case.

I'm sorry, but that is nonsense. Regardless of where this integration arises, what is inside of it is simply a mathematical function, and the result of the integration is also a mathematical function.

Other than that unknown constant of integration, this integral has only one solution. It does not matter what methods you used to arrive at that solution, the solution itself is the same function. If that solution is "inappropriate", you are out-of-luck. It is the solution. You can't change it.

Now some ways of expressing the solution are more useful than others, in that they allow you to calculate or approximate or manipulate it more easily or more accurately. But generally, you want to find the solution any way you can, then transform it into the most useful form.

So solve it by integration by parts, without assuming that each intermediary step must have some obvious physical meaning. Once you have a solution, figure out how to put it in the form you want.

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