## hartnn 3 years ago Efficient (or any) way to integrate this: $$\large \sqrt{a^2cos^2 x+b^2sin^2x}dx$$

1. hartnn

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

2. hartnn

@sauravshakya @siddhantsharan

3. hartnn

@Zekarias

4. AravindG

did you try competing square?

5. hartnn

adding and subtracting 2ab sinxcosx ?

6. AravindG

yep thats what came into my ind first not foolproof

7. hartnn

plz.,try...

8. AravindG

did that work @hartnn

9. Zekarias

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

10. hartnn

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

11. mukushla

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

12. mukushla

maybe

13. hartnn

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

14. hartnn

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

15. estudier

You may try the Weierstrass substitution....

16. mukushla

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

17. hartnn

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

18. mukushla

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

19. hartnn

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