Efficient (or any) way to integrate this:
$\large \sqrt{a^2cos^2 x+b^2sin^2x}dx$

Question

Efficient (or any) way to integrate this:
$\large \sqrt{a^2cos^2 x+b^2sin^2x}dx$

hartnn · Answer

$\huge \int \large \sqrt{a^2cos^2 x+b^2sin^2x}dx$

hartnn · Answer

@sauravshakya @siddhantsharan

hartnn · Answer

@Zekarias

aravindg · Answer

did you try competing square?

hartnn · Answer

adding and subtracting 2ab sinxcosx ?

aravindg · Answer

yep thats what came into my ind first not foolproof

hartnn · Answer

plz.,try...

aravindg · Answer

did that work @hartnn

anonymous · Answer

what if we insert 1-sin^2(x) instead of cos^2(x)

hartnn · Answer

$\sqrt{(acos x+bsinx)^2-2absinxcosx}$
hmm....what next ?

anonymous · Answer

seems to be an elliptic integral and there is no closed form for elliptic integrals

anonymous · Answer

maybe

hartnn · Answer

yes, i heard same thing earlier....but could not digest that this simple function is not integrable......

hartnn · Answer

no closed form...any *open* form ?

anonymous · Answer

You may try the Weierstrass substitution....

anonymous · Answer

more simpler$$\int \sqrt{1+\sin^2 x} \ \text{d}x$$but no closed form :)

hartnn · Answer

ohh...okk....so no closed form means?
do we write it in some new function form ?

anonymous · Answer

closed form means there is no antiderivative for integrand in terms of elementary functions

hartnn · Answer

ok,then.....no use banging our heads,solving this....m closing it.....