Is the Given Function an Even Function?

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In summary, the given function is an even function and the fact that it is even is only relevant when the limits of integration are symmetric, with a = -b. The student has tried using properties and substitutions, but has not found a satisfying result. The tutor advises to look carefully at the results of the substitution and suggests trying u=1/x.
  • #1
zorro
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Homework Statement


gif.latex?\textup{Evaluate}&space;\&space;\int_{\frac{1}{2}}^{2}\frac{sin(x-\frac{1}{x})}{x}dx.gif



The given function is an even function. I tried using properties but did not get any satisfying result.
 
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  • #2
Have you tried a substitution?
 
  • #3
The fact that the function is even is relevant only when the limits of integration are symmetric, with a = -b.
 
  • #4
LeonhardEuler said:
Have you tried a substitution?

I tried many things :biggrin:
But did not get any solution.

Mark44 said:
The fact that the function is even is relevant only when the limits of integration are symmetric, with a = -b.

Yes I forgot about that. Which property shall I use then?
 
  • #5
What substitutions did you try? If you look carefully at what you get from the substitution, the answer might pop out even though it looks like you've reached a dead end.
 
  • #6
LeonhardEuler said:
What substitutions did you try? If you look carefully at what you get from the substitution, the answer might pop out even though it looks like you've reached a dead end.

I looked carefully. Nothing popped out.
Now can you tell me what is that substitution?
 
  • #7
Try u=1/x
 
  • #8
heck that is easy!
Thanks! I got it as 0.
 

1. What is D.I.?

D.I. stands for "defect-induced". It refers to a problem or defect that arises due to the properties of a material, such as its structure or composition.

2. What are some common properties of D.I.?

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5. Can D.I. properties be beneficial?

Yes, in some cases, D.I. properties can be beneficial. For example, defects in semiconductors can be intentionally introduced to create desirable electronic properties for specific applications. Additionally, D.I. properties can also be useful in understanding material behavior and developing new materials with improved properties.

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