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Integrate $$\int(1+\cos^2(x))^{1/2} \, \mathrm{d} x.$$ I have tried applying substitution but it turns out to be really complex. By parts doesn't work either.

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    $\begingroup$ The integrand is $\sqrt{2}\sqrt{1-\tfrac{1}{2}\sin^2x}$. The rest of the story is here: en.wikipedia.org/wiki/… $\endgroup$ – J.G. Jul 3 '16 at 15:44
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    $\begingroup$ There is no elementary anti-derivative. $\endgroup$ – Zain Patel Jul 3 '16 at 15:50
  • $\begingroup$ Check that you don't have a copy error. If the + is changed to a - in your integrand, you'd have a rather easy integral! $\endgroup$ – John Molokach Jul 3 '16 at 16:54
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This integral is indeed in the form of an Elliptic integral of second kind (link). Elliptic integral of second kind $E(\phi,k)$ can be expressed in following form

$$E(\phi,k)=\int_0^\phi \sqrt{ 1-k^2 \sin^2 \theta}\: \mathrm{d}\theta.$$

Assuming $\; \sin^2 \theta = 1-\cos^2 \theta$, you can write your equation as follows

$$\int { \sqrt { 1+\cos ^{ 2 }{ \theta } } \mathrm{d}\theta } = \sqrt{2} \int { \sqrt { 1-\frac{1}{2}\sin ^{ 2 }{ \theta } } \mathrm{d}\theta }$$

If you set your integral to be definite integral in $[0, \phi]$, you will get $\sqrt{2}E(\phi,\sqrt{\frac{1}{2}})$ as the answer to your integral, else, I am not aware of other solution.

Please note that Elliptic integrals in general can not be represented using elementary function.

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