Evaluating a definite integral when a condition is given

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Homework Help Overview

The discussion revolves around evaluating a definite integral of the function f(x) under the condition that x²f(x) + f(1/x) = 0. The integral in question is from 0.6 to 1.5.

Discussion Character

  • Exploratory, Assumption checking

Approaches and Questions Raised

  • Participants discuss attempts to replace f(x) using the given equation and explore the implications of substituting y = 1/x in the integral. Questions arise about the correctness of the integral bounds and the necessity of additional information about f(x).

Discussion Status

The discussion is ongoing, with participants providing guidance on substitution methods and questioning the bounds of the integral. There is recognition that the problem may not be solvable without further information about the function f(x).

Contextual Notes

There is a noted concern regarding the lower limit of the integral, with some participants suggesting that if the limit is 0.6666... (or 2/3), the problem may be approached differently.

justwild
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Homework Statement


Given that x^{2}f(x)+f(\frac{1}{x})=0, then evaluate \int^{1.5}_{0.6}f(x)dx

Homework Equations


The Attempt at a Solution



tried to replace f(x) using the provided equation...didn't help
 
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Can you elaborate on how you tried replacing f(x)?
 
Have you looked at what happens if you substitute y = 1/x in the integral and the use the equation to substitute for f(1/y)?
(Are you sure you've quoted the bounds correctly? It's not from a lower bound of 0.66666... by any chance?)
 
haruspex said:
Have you looked at what happens if you substitute y = 1/x in the integral and the use the equation to substitute for f(1/y)?

well If I do that I shall be returning to the same problem statement...
 
justwild said:
well If I do that I shall be returning to the same problem statement...

If the lower limit in the integral is 0.6, you cannot answer the question without knowing more about the function f(x). If the lower limit is 0.6666... = 2/3, you can answer the question without knowing more about f(x).

The substitution u = 1/x DOES work if you do it judiciously!
 

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