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.

  • 1
    $\begingroup$ 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
  • 1
    $\begingroup$ 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

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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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