How to integrate 1/(1-A*cosx)^3?

In summary, the author is looking for a solution to integrate a function f(x) that is written in the form 1/(1-A*cosx)^3, but they don't know how to start. They found an answer on Wolfram Alpha, but they are unsure if they copied the integral correctly or if it is something more advanced.
  • #1
Arash.
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0
Hello every body.
I'm searching for a solution to integrate the function below but i couldn't find anything suitable yet.I don't know even how to start it !

can anybody help me finding the integral?
f(x)=1/(1-A*cosx)^3

Thanks you ...
 
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  • #2
Simplify the denominator with a substitution. Try the obvious substitution.
 
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  • #4
Ok. Judging from Emmanuel_Euler's Mathematica solution, evaluating the integral you wrote out above is certainly not a freshman exercise. Are you sure that you have copied out your integral correctly? Or is this something that arose in more advanced work?
 
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  • #5
Is this homework? If so, then see https://www.physicsforums.com/help/homeworkhelp/ .

One way to approach this would be to express the cosine using complex exponentials, then use the obvious substitution to get the integral into the form [itex]\int du/P(u)[/itex], where P is a polynomial. This might then be doable using partial fractions.
 
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  • #6
MarcusAgrippa said:
Ok. Judging from Emmanuel_Euler's Mathematica solution, evaluating the integral you wrote out above is certainly not a freshman exercise. Are you sure that you have copied out your integral correctly? Or is this something that arose in more advanced work?
yes,i copied correctly. why??
 
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  • #7
Emmanuel_Euler said:
yes,i copied correctly. why??
I believe the question was intended for the OP.
 
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  • #8
Thanks you every body ,
But yet i couldn't find the right way , and the integral i wrote is correct.
wolframalpha also doesn't show step by step guide.
 
  • #9
Arash. said:
wolframalpha also doesn't show step by step guide.
It does usually, but with this one it seems to take forever to load.
Anyway, you saw the answer, are you comfortable with that sort of an answer ?
[EDIT:- Ah, I see what you mean, after loading it said the solution was not avaliable, this can happen sometimes with hard integrals like the one you provided.]
 
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FAQ: How to integrate 1/(1-A*cosx)^3?

What is the general formula for integrating 1/(1-A*cosx)^3?

The general formula for integrating 1/(1-A*cosx)^3 is:
∫ 1/(1-A*cosx)^3 dx = (1/2)tan(x/2) / (1-A*cosx)^2 + (A/4) ∫ tan(x/2) / (1-A*cosx) dx

How can I simplify the integral of 1/(1-A*cosx)^3?

You can simplify the integral of 1/(1-A*cosx)^3 by using the trigonometric identity:
tan(x/2) = sin(x) / (1+cosx)
This will reduce the integral to:
∫ sin(x) / (1-A*cosx)^2 dx

What substitution should I use to solve the integral of 1/(1-A*cosx)^3?

The substitution u = tan(x/2) is commonly used to solve the integral of 1/(1-A*cosx)^3. This will transform the integral into a simpler form, as mentioned in the previous answer.

Can the integral of 1/(1-A*cosx)^3 be solved using partial fractions?

Yes, the integral of 1/(1-A*cosx)^3 can be solved using partial fractions. However, this method may require more complex calculations compared to using the substitution u = tan(x/2).

Are there any special cases to consider when integrating 1/(1-A*cosx)^3?

Yes, when A = 1, the integral of 1/(1-A*cosx)^3 becomes a special case. In this case, the integral can be solved using the substitution u = tan(x/2) and simplifying using the identity:
tan(x/2) = sin(x) / (1+cosx)
The resulting integral will be in the form of ∫ sin(x) / (1-cosx)^2 dx, which can be solved using another substitution.

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