Integral of a function gives me acrtan(2cot(x)). How to evaluate this?

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Discussion Overview

The discussion revolves around the evaluation of the integral $$\int \frac{(sin^22x)}{(4(1+3cos^2(x)))}dx$$ and its relationship to the expression $$\frac{5 x}{18} + \frac{(2 arctan(2 cot(x)))}{9} - \frac{sin(2 x)}{12}$$. Participants explore the implications of evaluating this integral over the interval [0, 60] and the physical interpretation of the results.

Discussion Character

  • Mathematical reasoning
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant presents the integral and its purported evaluation, expressing confusion about the behavior of the cotangent function at the lower limit of integration.
  • Another participant notes the importance of including a constant in the evaluation, suggesting that the original poster has identified this constant.
  • There is a question regarding the meaning and units of the interval [0, 60], indicating a need for clarification on the context of the integration.
  • A later reply emphasizes that the original poster had already indicated the integration is with respect to time, suggesting that the units may be understood in that context.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the implications of the integral's evaluation, particularly regarding the behavior at the limits of integration and the physical interpretation of the results. Multiple viewpoints are expressed regarding the significance of the constant and the units of the interval.

Contextual Notes

There are unresolved questions about the physical context of the integral, particularly concerning the interpretation of energy generation at time = 0 and the implications of the cotangent function approaching infinity.

Who May Find This Useful

Readers interested in integral calculus, particularly in the context of physics applications, may find this discussion relevant.

anj16
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Function is context is:
$$\int \frac{(sin^22x)}{(4(1+3cos^2(x)))}dx$$

And according to wolframalpha (I wasn't able to integrate this by myself) this integral is equal to

$$\frac{5 x}{18} + \frac{(2 arctan(2 cot(x)))}{9} - \frac{sin(2 x)}{12}$$

The above integral is the integral of a power function with respect to time.
And I wanted to evaluate this between [0,60]; but the problem is when evaluating the cot(x) at 0 this give me infinity which make the fraction in the middle ∏/9. But according to the physics of the equation there should not be any energy generated at time = 0 as nothing is happening at time 0. so what is going on??

Thanks!
 
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Plus a constant... and what you've done is figure out what that constant is
 
What does [0,60] represent? Does it have units?
 
SteamKing said:
What does [0,60] represent? Does it have units?

Not trying to be rude but in my original post I said "integrating with respect to time" so I didn't type time again; I thought it would be unnecessary.
 
Office_Shredder said:
Plus a constant... and what you've done is figure out what that constant is

That makes sense. I completely forgot about the constant. Thanks!
 

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