Definite integral

This is not really a homework.

Homework Statement

I am trying to solve one definite integral

Homework Equations

$$\int_0^{2 \pi} \frac{\sin^2{t}}{\sqrt{a\cos{t} + b}} dt$$

where $$a, b$$ are some positive numbers.

The Attempt at a Solution

I tried integrate by parts, also differentiate by parameter $$a$$. 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

$$\int_0^{2 \pi} \sqrt{a\cos{t} + b} \cos{t} dt$$

or

$$\int_0^{2 \pi} \frac{1}{\sqrt{a\cos{t} + b}} dt$$

cronxeh
Gold Member
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:
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.

Last edited by a moderator:
cronxeh
Gold Member
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