$$\int \frac{1}{\sqrt{5+4\cos x}}\,dx$$

I tried a lot to solve this simple looking problem. I tried $\cos x=2\cos ^2 {x\over 2}-1$ but no use. Tried half angle substitution of $\tan {x\over 2 }$ but ended up making the problem complex.

Need help. Thanks in advance.

  • 1
    $\begingroup$ You tried $\frac{\tan x}{2}$? Not $\tan\frac{x}{2}$? $\endgroup$ – J.G. May 2 '19 at 5:41
  • $\begingroup$ Sorry for the typo, actually I tried $\tan {x \over 2}$ $\endgroup$ – user585765 May 2 '19 at 5:46
  • $\begingroup$ What is elliptic integral? I came across this integral while solving a physics numerical. $\endgroup$ – user585765 May 2 '19 at 5:49
  • $\begingroup$ The limits were from 0 to $π/2$ $\endgroup$ – user585765 May 2 '19 at 5:49

Sometimes, mathematicians invent a name for the solution to one problem, then write the solutions to others in terms of that. For this problem, we need the incomplete elliptic integral of the first kind,$$F(x|m):=\int_0^x\frac{d\phi}{\sqrt{1-m\sin^2\phi}}.$$Since$$\int_0^x\frac{dt}{\sqrt{5+4\cos t}}=\int_0^x\frac{dt}{\sqrt{9-8\sin^2\frac{t}{2}}}=\frac23\int_0^{x/2}\frac{d\phi}{\sqrt{1-\frac89\sin^2\phi}}=\frac23 F\left(\left.\frac{x}{2}\right|\frac89\right),$$your original indefinite integral is $\frac23 F\left(\left.\frac{x}{2}\right|\frac89\right)+C$.


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