Definite integral

  • Thread starter zeebek
  • Start date
  • #1
27
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This is not really a homework.

Homework Statement



I am trying to solve one definite integral

Homework Equations



[tex] \int_0^{2 \pi} \frac{\sin^2{t}}{\sqrt{a\cos{t} + b}} dt[/tex]

where [tex]a, b[/tex] are some positive numbers.

The Attempt at a Solution



I tried integrate by parts, also differentiate by parameter [tex]a[/tex]. Does not help because of the square root. In this case I got some diff.equation for the integral I want, but I need to solve even more nasty integrals:

either

[tex] \int_0^{2 \pi} \sqrt{a\cos{t} + b} \cos{t} dt[/tex]

or

[tex] \int_0^{2 \pi} \frac{1}{\sqrt{a\cos{t} + b}} dt[/tex]
 

Answers and Replies

  • #2
cronxeh
Gold Member
974
10
You joking, right? Even by breaking this up into two separate integrals, the first one 1/sqrt(a*cos(t)+b), t=0..2pi and second one -cos^2(t)/sqrt(a*cos(t)+b), t=0..2pi.. This gives some complete elliptic integral of first one for first part, and series expansion for second part. At best you can get a complete series expansion of this with http://www.wolframalpha.com/input/?i=Expand+(1-cos^2(t))/sqrt(a*cos(t)+b)" and numerically evaluate it if you know a and b.
 
Last edited by a moderator:
  • #3
27
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You joking, right? Even by breaking this up into two separate integrals, the first one 1/sqrt(a*cos(t)+b), t=0..2pi and second one -cos^2(t)/sqrt(a*cos(t)+b), t=0..2pi.. This gives some complete elliptic integral of first one for first part, and series expansion for second part. At best you can get a complete series expansion of this with http://www.wolframalpha.com/input/?i=Expand+(1-cos^2(t))/sqrt(a*cos(t)+b)" and numerically evaluate it if you know a and b.

Thank you for your reply. As I said this is not a homework problem. In fact if somebody could help me represent this intergral via some generalized functions this would do. Sorry for the misconception.
 
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  • #4
cronxeh
Gold Member
974
10
Thank you for your reply. As I said this is not a homework problem. In fact if somebody could help me represent this intergral via some generalized functions this would do. Sorry for the misconception.

I can't see how this could be done. a and b make this impossible
 

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