Integral $\int\frac{dx}{\sqrt{\cos x + c}}$

Question

Could you help to find the answer of this integral?

$$\int\frac{dx}{\sqrt{\cos x + c}}$$

Does it have a solution? if not, how to prove it? Thank you,

https://www.integral-calculator.com/#expr=1%2Fsqrt%28cosx%2Bc%29, just click "Go!" and "Show steps". — thesmallprint, Oct 30 '17 at 18:14
You should take the to learn Mathjax The mark-up system for this site it avoids any confusion regarding what the equation actually is. — Warren Hill, Oct 30 '17 at 18:15

score 0 · Accepted Answer · answered Oct 30 '17 at 18:22

0

The solution is given by:

$${\frac {2}{c-1}J^{-1}\left( x/2,{\frac {\sqrt {2}}{\sqrt {c +1}}} \right) } $$

in which

$$J^{-1}(\phi,k)=\int_{0}^{\phi}\dfrac{du}{\sqrt{1-k^2\sin^2 u}}.$$

Hence, the integral does not seem to have a simple closed form solution.

answered Oct 30 '17 at 18:22

MrYouMath

15,833

Is there a name for this special function? It looks familiar. Edit: It's an elliptic integral – Dylan Oct 30 '17 at 18:29

score 0 · Answer 2 · answered Oct 30 '17 at 18:29

In a sense it does have a solution, $2F(\frac{x+c}{2}|2)+const$. The function $F$ is an elliptic integral of first kind, which is defined as a solution to this kind of integrals, and cannot be expressed in terms of elementary functions. Proving ideas are discussed in How can you prove that a function has no closed form integral?

Integral $\int\frac{dx}{\sqrt{\cos x + c}}$

2 Answers2