Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I want to find a symbolic expression for the following integral as a function of $f$ and $g$:

$$ \int_{0}^{\pi} \sqrt{1-\frac{2}{f + g \cdot \cos \theta}} \, d\theta $$

It is guaranteed that $f$ and $g$ are real numbers such that the argument to the square root is non-negative over the integration interval, e.g. $f = 36, g = -16$.

Unfortunately, I am stuck here. I can find solutions for $g = 0$, but other cases elude me.

Numerical integration handily yields numerical solutions that I have verified to be correct against the problem I'm trying to solve, but I'd appreciate a closed-form answer. Any help or insight on this would be greatly appreciated!

share|cite|improve this question
An elliptic integral. Plugging it into Maple, I get something in terms of elliptic integrals $K$, $F$, and $\Pi$. – GEdgar Jul 9 '12 at 13:24
Having a trigonometric function within a square root almost always leads to an elliptic integral... – J. M. Jul 9 '12 at 16:36

For example, Maple says $$ \int_{0}^{\pi} \sqrt{1 - \frac{2}{4 + \operatorname{cos} (t)}} d t = \frac{2}{3} K \Bigl(\frac{2}{3}\Bigr) - \frac{4 i}{\sqrt5} \lim_{z\to+\infty} \Pi \biggl(i z,3,\frac{3}{\sqrt5}\biggr) $$

share|cite|improve this answer

Maybe you could start at $$\int\sqrt{1-\frac{2}{1\pm\cos\left(x\right)}}\mathrm{d}x=2\mathop{\mathrm{atan_2}}\left(\sin\left(x\right),\cos\left(x\right)\pm1\right)-x$$ to reach you solution.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.